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is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same
2739:
Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values â the probability mass function (PMF) â or for sets of values, including infinite sets. For
3949:
An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to
2735:
Consider an experiment where a person is chosen at random. An example of a random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to their height. Associated with the random variable is a probability distribution that
10268:
8012:
3364:
1490:
A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are
8556:
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4496:
3950:
map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval is
5725:
The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of
6789:{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq h(y))=F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq h(y))=1-F_{X}(h(y)),&{\text{if }}h=g^{-1}{\text{ decreasing}}.\end{cases}}}
9625:
3970:
is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set.
2740:
example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum
6067:
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function
2822:
random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
2736:
allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
8522:
1035:
5195:(those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
584:
is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup.
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1205:, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous.
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8824:{\displaystyle {\begin{aligned}F_{Y}(y)&=1-F_{X}(-\log(e^{y}-1))\\&=1-{\frac {1}{(1+e^{\log(e^{y}-1)})^{\theta }}}\\&=1-{\frac {1}{(1+e^{y}-1)^{\theta }}}\\&=1-e^{-y\theta }.\end{aligned}}}
4320:
7396:
6230:
1978:
848:
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generates the Borel Ď-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that
5492:
5032:
4619:
An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails,
3559:
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers
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on the set of all possible values of the random variable. It is possible for two random variables to have identical distributions but to differ in significant ways; for instance, they may be
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There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
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10663:{\displaystyle f_{Y}(y)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}{\frac {1}{2{\sqrt {y}}}}(e^{-({\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}+e^{-(-{\sqrt {y}}-\mu )^{2}/(2\sigma ^{2})}).}
8358:
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from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable
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This graph shows how random variable is a function from all possible outcomes to real values. It also shows how random variable is used for defining probability mass functions.
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of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a
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10032:(one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
6853:
10996:
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
4680:, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or
341:
9204:
10263:{\displaystyle f_{Y}(y)=f_{X}(g_{1}^{-1}(y))\left|{\frac {dg_{1}^{-1}(y)}{dy}}\right|+f_{X}(g_{2}^{-1}(y))\left|{\frac {dg_{2}^{-1}(y)}{dy}}\right|.}
4644:
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see
10354:
4238:
632:
13354:
8007:{\displaystyle \operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),}
2718:
that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on
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11384:
378:
events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical
11137:
8163:
3359:{\displaystyle Y(\omega )={\begin{cases}1,&{\text{if }}\omega ={\text{heads}},\\0,&{\text{if }}\omega ={\text{tails}}.\end{cases}}}
14009:
9401:
4056:
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5633:
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is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as
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10866:
To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal
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3580:
13846:
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can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of
1941:
811:
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that this random variable will have the value â1. Other ranges of values would have half the probabilities of the last example.
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4676:
to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the
3525:{\displaystyle f_{Y}(y)={\begin{cases}{\tfrac {1}{2}},&{\text{if }}y=1,\\{\tfrac {1}{2}},&{\text{if }}y=0,\end{cases}}}
3202:
334:
6321:
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4491:{\displaystyle f_{X}(x)={\begin{cases}\displaystyle {1 \over b-a},&a\leq x\leq b\\0,&{\text{otherwise}}.\end{cases}}}
3939:
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12269:
11969:
4965:
9620:{\displaystyle f_{Y}(y)=2{\frac {1}{\sqrt {2\pi }}}e^{-y/2}{\frac {1}{2{\sqrt {y}}}}={\frac {1}{\sqrt {2\pi y}}}e^{-y/2}.}
4648:. The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
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If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum
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have the same distribution. This provides, for example, a useful method of checking equality of certain functions of
2669:
1907:
1142:
621:
327:
315:
274:
10952:
4612:. It can be realized as a mixture of a discrete random variable and a continuous random variable; in which case the
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13834:
13708:
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3840:
2230:
1261:
1209:
17:
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This notion is typically the least useful in probability theory because in practice and in theory, the underlying
5151:
represents the set of values that the random variable can take (such as the set of real numbers), and a member of
2003:
655:
is taken to be automatically valued in the real numbers, with more general random quantities instead being called
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of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the
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of two or more random variables on the same probability space. In practice, one often disposes of the space
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9198:(one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
8517:{\displaystyle F_{Y}(y)=P(Y\leq y)=P(\mathrm {log} (1+e^{-X})\leq y)=P(X\geq -\mathrm {log} (e^{y}-1)).\,}
4826:
1212:, which describes the probability that the random variable will be less than or equal to a certain value.
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11994:
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1423:
1030:{\displaystyle \operatorname {P} (X\in S)=\operatorname {P} (\{\omega \in \Omega \mid X(\omega )\in S\})}
572:
Informally, randomness typically represents some fundamental element of chance, such as in the roll of a
11713:
10727:
In increasing order of strength, the precise definition of these notions of equivalence is given below.
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5773:
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4672:) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a
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13681:
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12945:
12763:
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12118:
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10867:
7606:
5255:
2665:
109:
11473:
George Mackey (July 1980). "Harmonic analysis as the exploitation of symmetry â a historical survey".
7747:
5852:. Once the "average value" is known, one could then ask how far from this average value the values of
3093:
3025:
1252:). In this case, the structure of the real numbers makes it possible to define quantities such as the
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12878:
12748:
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11217:
A significant theme in mathematical statistics consists of obtaining convergence results for certain
4689:
4669:
3825:{\displaystyle f_{X}(S)={\frac {\min(S-1,13-S)}{36}},{\text{ for }}S\in \{2,3,4,5,6,7,8,9,10,11,12\}}
3552:
3377:
2828:
1146:
597:
225:
11575:
Dekking, Frederik Michel; Kraaikamp, Cornelis; Lopuhaä, Hendrik Paul; Meester, Ludolf Erwin (2005).
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5228:
5154:
5090:
4892:
4418:
1845:
are ordinary real-valued random variables provided that the function is real-valued. For example, a
1458:, using one or more real numbers. In this case, a random element may optionally be represented as a
1227:
1073:
14181:
13948:
13811:
13496:
13461:
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12561:
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8839:
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628:
284:
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168:
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8267:
5387:
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922:
635:, corresponding to whether a random variable is valued in a countable subset or in an interval of
14090:
13703:
13643:
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13202:
12940:
12802:
12792:
12642:
12556:
11359:
7484:
4325:
1315:
263:
134:
10856:{\displaystyle \operatorname {P} (X\leq x)=\operatorname {P} (Y\leq x)\quad {\text{for all }}x.}
9978:{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}
9144:{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|.}
7038:
4160:
2052:
14128:
14058:
13851:
13788:
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12009:
9789:{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-(x-\mu )^{2}/(2\sigma ^{2})}.}
8122:{\displaystyle F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.}
7221:{\displaystyle f_{Y}(y)=\sum _{i}f_{X}(g_{i}^{-1}(y))\left|{\frac {dg_{i}^{-1}(y)}{dy}}\right|}
5962:
5565:{\displaystyle \{\omega :X(\omega )\leq r\}\in {\mathcal {F}}\qquad \forall r\in \mathbb {R} .}
3975:
3923:
3191:, one gets a discrete function that is not necessarily a step function (piecewise constant).
3161:
2807:{\displaystyle \operatorname {PMF} (0)+\operatorname {PMF} (2)+\operatorname {PMF} (4)+\cdots }
2169:. The probability distribution "forgets" about the particular probability space used to define
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1170:
379:
158:
9802:
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6416:
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484:
14153:
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11989:
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7861:
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admits at most a countable number of roots (i.e., a finite, or countably infinite, number of
5070:
4381:
3943:
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3135:
2679:
2647:
2376:
1465:
1198:
861:
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741:
401:
391:
299:
258:
163:
129:
11093:{\displaystyle d_{\infty }(X,Y)=\operatorname {ess} \sup _{\omega }|X(\omega )-Y(\omega )|,}
10874:. However, the moment generating function exists only for distributions that have a defined
7721:
7401:
6259:
6071:
3067:
2880:
14076:
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13576:
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5765:
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1265:
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289:
183:
76:
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8:
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11444:. A. Aldo Faisal, Cheng Soon Ong. Cambridge, United Kingdom: Cambridge University Press.
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10707:, as these do not preserve non-negativity or total integral 1âbut they are closed under
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14048:
13902:
13798:
13747:
13623:
13520:
13504:
13481:
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12992:
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12703:
12674:
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12457:
12143:
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11936:
11922:
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10708:
10704:
10015:
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9649:
9181:
9161:
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8140:
7588:
7560:
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7322:
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6991:
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2192:
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1983:
1846:
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124:
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10875:
6958:{\displaystyle f_{Y}(y)=f_{X}{\bigl (}h(y){\bigr )}\left|{\frac {dh(y)}{dy}}\right|.}
6820:
5219:
4741:
4665:
4557:
3927:
2719:
806:
669:
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294:
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99:
4130:
depends only on the length of the subinterval. This implies that the probability of
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11584:
10690:
The probability distribution of the sum of two independent random variables is the
6827:
can be found by differentiating both sides of the above expression with respect to
6490:
5064:
in the target space by looking at its preimage, which by assumption is measurable.
4781:
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2641:
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1431:
1419:
1272:
763:
605:
564:
478:
119:
49:
4631:= the value of the spinner as in the preceding example. There is a probability of
3960:. Instead of speaking of a probability mass function, we say that the probability
1434:, where one is often interested in modeling the random variation of non-numerical
14085:
13829:
13691:
13618:
13293:
13167:
13140:
13117:
13086:
12713:
12708:
12662:
12392:
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11868:
11862:
11840:
11834:
11821:
11807:
11801:
11797:
11777:
11756:
11683:. Tsitsiklis, John N., ΤĎΚĎĎΚκΝΎĎ, ÎÎšÎŹÎ˝Î˝ÎˇĎ Î. Belmont, Mass.: Athena Scientific.
11349:
8527:
The last expression can be calculated in terms of the cumulative distribution of
6150:
1745:
1355:
1335:
648:
644:
146:
13575:
9324:{\displaystyle f_{Y}(y)=2f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.}
14034:
14029:
12492:
12422:
12068:
11364:
11354:
11344:
11108:
6174:
5727:
4661:
2330:
1435:
1411:
1253:
1099:
657:
589:
210:
11459:
14277:
14191:
14158:
14021:
13982:
13793:
13762:
13226:
13180:
12785:
12487:
12314:
12078:
12073:
11606:
11369:
11196:
10897:
10712:
10439:{\displaystyle {\frac {dg_{1,2}^{-1}(y)}{dy}}=\pm {\frac {1}{2{\sqrt {y}}}}.}
4673:
4499:
3199:
The possible outcomes for one coin toss can be described by the sample space
1850:
1459:
1331:
1327:
1134:
672:
was the first person "to think systematically in terms of random variables".
665:
83:
11698:
3922:) but there is a positive probability that its value will lie in particular
2644:
in the case of discrete random variables). The underlying probability space
14133:
14066:
14043:
13958:
13288:
12584:
12482:
12417:
12359:
12344:
12281:
12236:
11439:
11309:
10700:
9835:
We can find the density using the above formula for a change of variables:
9001:
We can find the density using the above formula for a change of variables:
6112:
3856:
2818:
1895:
is a random function on any set (typically time, space, or a discrete set).
1892:
1688:
1438:. In some cases, it is nonetheless convenient to represent each element of
1415:
855:
395:
310:
220:
104:
11588:
5223:
3539:
14176:
14138:
13821:
13722:
13584:
13397:
13364:
12856:
12773:
12768:
12412:
12369:
12349:
12329:
12319:
12088:
10692:
6233:
5285:, which is the Ď-algebra generated by the collection of all open sets in
4127:
1221:
1220:
The term "random variable" in statistics is traditionally limited to the
1067:
636:
557:
371:
230:
71:
59:
11576:
5418:
In this case the observation space is the set of real numbers. Recall,
3851:", which would correspond to numbers which have a finite probability of
1980:
is given, we can ask questions like "How likely is it that the value of
13022:
12502:
12202:
12133:
12083:
12058:
11978:
11619:
11374:
639:. There are other important possibilities, especially in the theory of
375:
88:
34:
11623:
Introduction to
Probability and Stochastic Processes with Applications
7481:.) The same procedure that allowed one to go from a probability space
4126:
random variable" (CURV) if the probability that it takes a value in a
13175:
13027:
12647:
12442:
12354:
12339:
12334:
12299:
11185:{\displaystyle X(\omega )=Y(\omega )\qquad {\hbox{for all }}\omega .}
9989:
8254:{\displaystyle F_{X}(x)=P(X\leq x)={\frac {1}{(1+e^{-x})^{\theta }}}}
5460:
is the probability space. For a real observation space, the function
4657:
4616:
will be the weighted average of the CDFs of the component variables.
3551:
is a discrete random variable whose distribution is described by the
3369:
1361:
11537:
481:(e.g. corresponding to the domain above, the range might be the set
12691:
12309:
12186:
12181:
12176:
11218:
10943:
10735:
If the sample space is a subset of the real line, random variables
9472:{\displaystyle {\frac {dg^{-1}(y)}{dy}}={\frac {1}{2{\sqrt {y}}}}.}
5873:
4886:
3659:
3261:
that models a $ 1 payoff for a successful bet on heads as follows:
1351:
1257:
215:
11927:
4115:{\displaystyle X_{I}\sim \operatorname {U} (I)=\operatorname {U} }
3839:
Formally, a continuous random variable is a random variable whose
2125:
Recording all these probabilities of outputs of a random variable
2000:
is equal to 2?". This is the same as the probability of the event
556:
mapped to 1). Typically, the range of a random variable is set of
14196:
13897:
11915:
Introduction to
Queueing Theory and Stochastic Teletraffic Models
5710:{\displaystyle \{\omega :X(\omega )\leq r\}=X^{-1}((-\infty ,r])}
3946:, or mixes of an absolutely continuous part and a singular part.
8955:{\displaystyle f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}.}
3158:. Taking for instance an enumeration of all rational numbers as
14118:
13099:
13073:
13053:
12304:
12095:
5575:
This definition is a special case of the above because the set
2189:
and only records the probabilities of various output values of
1294:
of values. Thus one can consider random elements of other sets
785:. The technical axiomatic definition requires the sample space
11574:
10942:) if, and only if, the probability that they are different is
11947:
5406:, then such a real-valued random variable is called simply a
4315:{\displaystyle \Pr \left(X_{I}\in \right)={\frac {d-c}{b-a}}}
1347:
11516:
Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003).
11203:
is rarely explicitly characterized or even characterizable.
7391:{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }
6225:{\displaystyle g\colon \mathbb {R} \rightarrow \mathbb {R} }
430:
which are the possible upper sides of a flipped coin heads
12038:
10872:
independent, identically distributed (IID) random variables
6782:
4588:, which are a unifying framework for all random variables.
4584:
distributed uniformly on the unit interval. This exploits
4484:
3518:
3352:
1973:{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}
843:{\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} )}
573:
11131:
if they are equal as functions on their measurable space:
6115:
values "red", "blue" or "green", the real-valued function
5111:
is a measurable subset of possible outcomes, the function
4498:
Of particular interest is the uniform distribution on the
11620:
L. CastaĂąeda; V. Arunachalam & S. Dharmaraja (2012).
7848:{\displaystyle F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.}
7708:{\displaystyle F_{Y}(y)=\operatorname {P} (X^{2}\leq y).}
3234:{\displaystyle \Omega =\{{\text{heads}},{\text{tails}}\}}
1149:
that assigns a probability to each value in the image of
6383:{\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).}
4613:
1106:) is used to denote a random variable not of this form.
11864:
Probability, Random
Variables, and Stochastic Processes
6497:, then the previous relation can be extended to obtain
5487:{\displaystyle X\colon \Omega \rightarrow \mathbb {R} }
1767:
may be represented as a collection of random variables
374:
formalization of a quantity or object which depends on
11259:
of random variables can converge to a random variable
11170:
8103:
7829:
6177:
and other moments of this function can be determined.
5131:
gives the probability of each such measurable subset,
4664:. Continuous random variables are defined in terms of
3983:
3869:
3482:
3444:
2959:
2916:
2883:
2831:
1462:(all defined on the same underlying probability space
11577:"A Modern Introduction to Probability and Statistics"
11265:
11238:
11140:
11005:
10955:
10907:
10795:
10753:
10458:
10357:
10282:
10041:
10018:
9998:
9844:
9805:
9679:
9652:
9491:
9404:
9343:
9207:
9184:
9164:
9010:
8971:
8883:
8856:
8559:
8533:
8361:
8296:
8270:
8166:
8143:
8026:
7893:
7864:
7801:
7750:
7724:
7651:
7615:
7591:
7563:
7519:
7487:
7463:
7439:
7404:
7368:
7345:
7325:
7298:
7237:
7090:
7041:
7014:
6994:
6974:
6856:
6833:
6805:
6506:
6475:
6455:
6419:
6399:
6324:
6301:
6262:
6242:
6202:
6159:
6121:
6074:
6050:
5998:
5965:
5945:
5918:
5886:
5858:
5817:
5776:
5736:
5636:
5581:
5503:
5466:
5424:
5390:
5370:
5348:
5311:
5291:
5258:
5231:
5204:
5181:
5157:
5137:
5117:
5093:
5073:
5040:
5027:{\displaystyle X^{-1}(B)=\{\omega :X(\omega )\in B\}}
4968:
4919:
4895:
4861:
4829:
4791:
4750:
4704:
4566:
4542:
4507:
4421:
4390:
4338:
4241:
4163:
4136:
4059:
3677:
3583:
3577:
given by the function that maps the pair to the sum:
3412:
3385:
3267:
3247:
3205:
3164:
3138:
3096:
3070:
3028:
2746:
2702:
2682:
2650:
2620:
2600:
2573:
2515:
2495:
2475:
2448:
2426:
2399:
2379:
2359:
2339:
2308:
2241:
2215:
2195:
2175:
2155:
2131:
2092:
2055:
2006:
1986:
1944:
1910:
1859:
1822:
1802:
1796:, giving the function's values at the various points
1773:
1753:
1737:
matrix of random variables, whose values specify the
1717:
1697:
1670:
1650:
1597:
1547:
1497:
1468:
1444:
1391:
1369:
1300:
1280:
1230:
1179:
1155:
1119:
1076:
1052:
954:
925:
905:
864:
814:
791:
771:
744:
712:
689:
542:
522:
487:
456:
436:
404:
13860:
Autoregressive conditional heteroskedasticity (ARCH)
11290:
8157:
is a random variable with a cumulative distribution
2288:{\displaystyle F_{X}(x)=\operatorname {P} (X\leq x)}
8290:is a fixed parameter. Consider the random variable
5935:Mathematically, this is known as the (generalised)
5034:. This definition enables us to measure any subset
4046:{\textstyle I==\{x\in \mathbb {R} :a\leq x\leq b\}}
3915:{\textstyle \forall c\in \mathbb {R} :\;\Pr(X=c)=0}
2640:in the case of continuous random variables, or the
1169:. If the image is uncountably infinite (usually an
616:of the random variable; the distribution is thus a
13322:
11271:
11251:
11184:
11092:
10985:
10935:{\displaystyle X\;{\stackrel {\text{a.s.}}{=}}\;Y}
10934:
10855:
10778:
10719:
10662:
10438:
10337:
10262:
10024:
10004:
9977:
9827:
9788:
9658:
9619:
9471:
9384:
9323:
9190:
9170:
9143:
8993:
8954:
8862:
8823:
8542:
8516:
8344:
8282:
8253:
8149:
8121:
8006:
7876:
7847:
7784:
7736:
7707:
7634:
7597:
7569:
7549:
7505:
7469:
7445:
7425:
7390:
7351:
7331:
7304:
7280:
7220:
7069:
7027:
7000:
6980:
6957:
6839:
6811:
6788:
6481:
6461:
6441:
6405:
6382:
6307:
6283:
6248:
6224:
6165:
6141:
6095:
6056:
6032:
5984:
5951:
5924:
5904:
5872:typically are, a question that is answered by the
5864:
5844:
5803:
5754:
5709:
5623:{\displaystyle \{(-\infty ,r]:r\in \mathbb {R} \}}
5622:
5564:
5486:
5452:
5398:
5376:
5354:
5333:
5297:
5277:
5241:
5210:
5187:
5167:
5143:
5123:
5103:
5079:
5056:
5026:
4954:
4905:
4877:
4847:
4813:
4772:
4732:
4646:Lebesgue's decomposition theorem § Refinement
4572:
4548:
4525:
4490:
4368:
4314:
4199:
4149:
4114:
4045:
3914:
3824:
3650:
3524:
3398:
3358:
3253:
3233:
3183:
3150:
3124:
3082:
3056:
3014:
2945:
2902:
2869:
2806:
2710:
2688:
2656:
2628:
2606:
2586:
2555:
2501:
2481:
2461:
2434:
2412:
2385:
2365:
2345:
2321:
2287:
2221:
2201:
2181:
2161:
2137:
2114:
2078:
2041:
1992:
1972:
1930:
1883:
1837:
1808:
1788:
1759:
1729:
1703:
1676:
1656:
1633:
1583:
1533:
1474:
1450:
1397:
1375:
1306:
1286:
1244:
1185:
1161:
1125:
1090:
1058:
1029:
937:
911:
888:
842:
797:
777:
750:
730:
695:
548:
528:
508:
462:
442:
422:
11714:"Economics 245A â Introduction to Measure Theory"
11515:
10338:{\displaystyle x=g_{1,2}^{-1}(y)=\pm {\sqrt {y}}}
3241:. We can introduce a real-valued random variable
2075:
2038:
1482:, which allows the different random variables to
854:). A random variable is often denoted by capital
851:
14275:
11041:
10786:) if they have the same distribution functions:
7453:, since the composition of measurable functions
6180:
4695:The measure-theoretic definition is as follows.
4242:
3888:
3703:
13408:Multivariate adaptive regression splines (MARS)
11412:
6173:has the value "green", 0 otherwise. Then, the
5413:
4651:
4586:properties of cumulative distribution functions
3015:{\textstyle F=\sum _{n}b_{n}\delta _{a_{n}}(x)}
1931:{\displaystyle X\colon \Omega \to \mathbb {R} }
1853:is a random function of some index set such as
1271:However, the definition above is valid for any
10986:{\displaystyle \operatorname {P} (X\neq Y)=0.}
5992:of functions such that the expectation values
3555:plotted as the height of picture columns here.
2229:is real-valued, can always be captured by its
1414:is particularly useful in disciplines such as
627:It is common to consider the special cases of
11963:
11475:Bulletin of the American Mathematical Society
11472:
11385:Relationships among probability distributions
11232:There are various senses in which a sequence
6910:
6891:
6799:With the same hypotheses of invertibility of
3930:. Continuous random variables usually admit
3834:
2469:is called the "(probability) distribution of
1641:and the position of the 1 indicates the word.
576:; it may also represent uncertainty, such as
335:
11754:
11737:
7457:. (However, this is not necessarily true if
5979:
5966:
5664:
5637:
5617:
5582:
5531:
5504:
5021:
4994:
4040:
4008:
3819:
3753:
3651:{\displaystyle X((n_{1},n_{2}))=n_{1}+n_{2}}
3228:
3212:
3178:
3165:
2864:
2851:
2845:
2832:
2042:{\displaystyle \{\omega :X(\omega )=2\}\,\!}
2034:
2007:
1208:Any random variable can be described by its
1021:
988:
503:
488:
417:
405:
11775:
11646:
11640:
10730:
8345:{\displaystyle Y=\mathrm {log} (1+e^{-X}).}
7292:. The formulas for densities do not demand
5175:is a "well-behaved" (measurable) subset of
4955:{\displaystyle X^{-1}(B)\in {\mathcal {F}}}
2730:
12008:
11970:
11956:
11832:
11796:
11437:
10928:
10911:
7557:can be used to obtain the distribution of
5453:{\displaystyle (\Omega ,{\mathcal {F}},P)}
4733:{\displaystyle (\Omega ,{\mathcal {F}},P)}
3887:
3022:is a discrete distribution function. Here
342:
328:
12621:
11926:
11719:. University of California, Santa Barbara
11678:
8513:
7524:
7384:
7376:
7077:) then the previous relation between the
6218:
6210:
5613:
5555:
5480:
5392:
4660:definition of a random variable involves
4324:where the last equality results from the
4018:
3880:
2704:
2622:
2428:
2074:
2037:
1924:
1899:
1238:
1201:, its distribution can be described by a
1084:
27:Variable representing a random phenomenon
11934:
11911:
11857:
11413:Blitzstein, Joe; Hwang, Jessica (2014).
11279:. These are explained in the article on
10715:of the space of functions (or measures).
5939:: for a given class of random variables
4369:{\displaystyle X\sim \operatorname {U} }
3942:; but some continuous distributions are
3934:(PDF), which characterize their CDF and
3855:. Instead, continuous random variables
3538:
563:
11758:A modern approach to probability theory
11755:Fristedt, Bert; Gray, Lawrence (1996).
11711:
10881:
9385:{\displaystyle x=g^{-1}(y)={\sqrt {y}}}
2594:with respect to some reference measure
1711:given vertices may be represented as a
1358:. One may then specifically refer to a
588:In the formal mathematical language of
470:as the result from tossing a coin); and
14:
14276:
13934:KaplanâMeier estimator (product limit)
11221:of random variables; for instance the
5222:, then the most common choice for the
3938:; such distributions are also called
2696:altogether and just puts a measure on
2636:(often, this reference measure is the
2209:. Such a probability distribution, if
1460:vector of real-valued random variables
633:absolutely continuous random variables
14007:
13574:
13321:
12620:
12390:
12007:
11951:
11674:
11672:
11670:
11668:
11330:Pairwise independent random variables
7550:{\displaystyle (\mathbb {R} ,dF_{X})}
7319:to probability, if a random variable
5845:{\displaystyle f(\operatorname {E} )}
5341:-valued random variable is called an
5067:In more intuitive terms, a member of
4855:, which means that, for every subset
4384:normalized by the interval's length:
3547:of the numbers on the two dice, then
1634:{\displaystyle (0\ 0\ 1\ 0\ \cdots )}
1584:{\displaystyle (0\ 1\ 0\ 0\ \cdots )}
1534:{\displaystyle (1\ 0\ 0\ 0\ \cdots )}
919:takes on a value in a measurable set
14244:
13944:Accelerated failure time (AFT) model
11705:
11651:(3rd ed.). Wiley. p. 187.
11509:
11408:
11406:
10779:{\displaystyle X{\stackrel {d}{=}}Y}
10699:Probability distributions are not a
5494:is a real-valued random variable if
4848:{\displaystyle X\colon \Omega \to E}
4556:can be generated by calculating the
2877:are countable sets of real numbers,
731:{\displaystyle X\colon \Omega \to E}
643:, wherein it is natural to consider
592:, a random variable is defined as a
14256:
13539:Analysis of variance (ANOVA, anova)
12391:
11396:
10675:noncentral chi-squared distribution
7281:{\displaystyle x_{i}=g_{i}^{-1}(y)}
5087:is a possible outcome, a member of
5057:{\displaystyle B\in {\mathcal {E}}}
4878:{\displaystyle B\in {\mathcal {E}}}
608:. This allows consideration of the
24:
13634:CochranâMantelâHaenszel statistics
12260:Pearson product-moment correlation
11665:
11119:Finally, the two random variables
11011:
10956:
10820:
10796:
10684:
8481:
8478:
8475:
8418:
8415:
8412:
8310:
8307:
8304:
7963:
7925:
7894:
7674:
7491:
7440:
7346:
6675:
6570:
6529:
6347:
6149:can be constructed; this uses the
6033:{\displaystyle \operatorname {E} }
5999:
5905:{\displaystyle \operatorname {E} }
5887:
5824:
5804:{\displaystyle \operatorname {E} }
5777:
5755:{\displaystyle \operatorname {E} }
5737:
5692:
5591:
5545:
5539:
5473:
5436:
5428:
5334:{\displaystyle (E,{\mathcal {E}})}
5323:
5261:
5234:
5160:
5096:
5074:
5049:
4947:
4898:
4870:
4836:
4814:{\displaystyle (E,{\mathcal {E}})}
4803:
4773:{\displaystyle (E,{\mathcal {E}})}
4762:
4716:
4708:
4573:{\displaystyle \operatorname {D} }
4567:
4549:{\displaystyle \operatorname {D} }
4543:
4345:
4091:
4073:
3870:
3662:) has a probability mass function
3206:
2683:
2651:
2556:{\displaystyle f_{X}=dp_{X}/d\mu }
2380:
2333:terms, we use the random variable
2264:
1964:
1956:
1948:
1917:
1664:may be represented as a vector of
1644:A random sentence of given length
1469:
1137:, the random variable is called a
997:
979:
955:
834:
826:
818:
792:
745:
719:
25:
14295:
11887:
11836:Foundations of Modern Probability
11466:
11403:
5278:{\displaystyle {\mathcal {B}}(E)}
1938:defined on the probability space
1197:. In the special case that it is
1143:discrete probability distribution
14255:
14243:
14231:
14218:
14217:
14008:
11441:Mathematics for machine learning
11293:
10012:has two corresponding values of
9178:has two corresponding values of
8836:cumulative distribution function
7785:{\displaystyle P(X^{2}\leq y)=0}
6968:If there is no invertibility of
6293:cumulative distribution function
4602:cumulative distribution function
4215:of the subinterval, that is, if
3841:cumulative distribution function
3125:{\displaystyle \delta _{t}(x)=1}
3057:{\displaystyle \delta _{t}(x)=0}
2870:{\textstyle \{a_{n}\},\{b_{n}\}}
2231:cumulative distribution function
1849:is a random function of time, a
1262:cumulative distribution function
1210:cumulative distribution function
1041:
48:
13893:Least-squares spectral analysis
11731:
11438:Deisenroth, Marc Peter (2020).
11281:convergence of random variables
11213:Convergence of random variables
11168:
11103:where "ess sup" represents the
10841:
10720:Equivalence of random variables
10696:of each of their distributions.
9988:In this case the change is not
9154:In this case the change is not
8109:
8101:
7835:
7827:
5544:
3859:take an exact prescribed value
2817:In examples such as these, the
1410:This more general concept of a
1103:
12874:Mean-unbiased minimum-variance
11977:
11679:Bertsekas, Dimitri P. (2002).
11613:
11568:
11544:
11485:
11431:
11206:
11165:
11159:
11150:
11144:
11083:
11079:
11073:
11064:
11058:
11051:
11028:
11016:
10974:
10962:
10838:
10826:
10814:
10802:
10654:
10649:
10633:
10619:
10599:
10583:
10567:
10553:
10536:
10525:
10475:
10469:
10394:
10388:
10319:
10313:
10273:The inverse transformation is
10239:
10233:
10202:
10199:
10193:
10172:
10141:
10135:
10104:
10101:
10095:
10074:
10058:
10052:
9954:
9948:
9917:
9914:
9908:
9887:
9861:
9855:
9778:
9762:
9748:
9735:
9696:
9690:
9508:
9502:
9430:
9424:
9369:
9363:
9334:The inverse transformation is
9300:
9294:
9268:
9265:
9259:
9243:
9224:
9218:
9120:
9114:
9083:
9080:
9074:
9053:
9027:
9021:
8900:
8894:
8770:
8744:
8710:
8704:
8685:
8665:
8640:
8637:
8618:
8606:
8580:
8574:
8507:
8504:
8485:
8462:
8453:
8444:
8422:
8408:
8399:
8387:
8378:
8372:
8336:
8314:
8239:
8216:
8204:
8192:
8183:
8177:
8098:
8085:
8069:
8059:
8043:
8037:
7998:
7969:
7957:
7943:
7935:
7931:
7919:
7900:
7818:
7812:
7773:
7754:
7699:
7680:
7668:
7662:
7544:
7520:
7500:
7488:
7420:
7414:
7380:
7275:
7269:
7200:
7194:
7163:
7160:
7154:
7133:
7107:
7101:
7064:
7051:
6934:
6928:
6905:
6899:
6873:
6867:
6739:
6736:
6730:
6724:
6702:
6699:
6693:
6681:
6628:
6625:
6619:
6613:
6597:
6594:
6588:
6576:
6556:
6547:
6541:
6535:
6523:
6517:
6374:
6365:
6359:
6353:
6341:
6335:
6278:
6272:
6214:
6136:
6122:
6084:
6078:
6027:
6024:
6018:
6005:
5899:
5893:
5839:
5836:
5830:
5821:
5798:
5795:
5789:
5783:
5749:
5743:
5730:of a random variable, denoted
5704:
5701:
5686:
5683:
5655:
5649:
5600:
5585:
5522:
5516:
5476:
5447:
5425:
5328:
5312:
5272:
5266:
5242:{\displaystyle {\mathcal {E}}}
5168:{\displaystyle {\mathcal {E}}}
5104:{\displaystyle {\mathcal {F}}}
5012:
5006:
4988:
4982:
4939:
4933:
4906:{\displaystyle {\mathcal {F}}}
4839:
4808:
4792:
4767:
4751:
4727:
4705:
4520:
4508:
4407:
4401:
4363:
4351:
4275:
4263:
4194:
4182:
4176:
4164:
4109:
4097:
4085:
4079:
4002:
3990:
3903:
3891:
3730:
3706:
3694:
3688:
3619:
3616:
3590:
3587:
3429:
3423:
3277:
3271:
3113:
3107:
3045:
3039:
3009:
3003:
2795:
2789:
2777:
2771:
2759:
2753:
2353:to "push-forward" the measure
2282:
2270:
2258:
2252:
2109:
2103:
2071:
2059:
2025:
2019:
1967:
1945:
1920:
1832:
1826:
1816:in the function's domain. The
1783:
1777:
1628:
1598:
1578:
1548:
1528:
1498:
1245:{\displaystyle E=\mathbb {R} }
1091:{\displaystyle E=\mathbb {R} }
1024:
1012:
1006:
985:
973:
961:
837:
815:
722:
115:Collectively exhaustive events
13:
1:
14187:Geographic information system
13403:Simultaneous equations models
11776:Billingsley, Patrick (1995).
11747:
11647:Billingsley, Patrick (1995).
11391:
9799:Consider the random variable
8965:Consider the random variable
7433:is also a random variable on
7079:probability density functions
6825:probability density functions
6181:Functions of random variables
4591:
3932:probability density functions
2946:{\textstyle \sum _{n}b_{n}=1}
1884:{\displaystyle 1,2,\ldots ,n}
1215:
1145:, i.e. can be described by a
1098:. In some contexts, the term
675:
582:interpretation of probability
13370:Coefficient of determination
12981:Uniformly most powerful test
11581:Springer Texts in Statistics
11325:Multivariate random variable
9666:is a random variable with a
9641:
8872:standard normal distribution
8870:is a random variable with a
8845:
8283:{\displaystyle \theta >0}
8132:
7580:
5414:Real-valued random variables
5399:{\displaystyle \mathbb {R} }
4652:Measure-theoretic definition
4330:probability density function
3534:
3194:
2711:{\displaystyle \mathbb {R} }
2629:{\displaystyle \mathbb {R} }
2435:{\displaystyle \mathbb {R} }
2300:probability density function
1203:probability density function
938:{\displaystyle S\subseteq E}
852:measure-theoretic definition
7:
13939:Proportional hazards models
13883:Spectral density estimation
13865:Vector autoregression (VAR)
13299:Maximum posterior estimator
12531:Randomized controlled trial
11900:Encyclopedia of Mathematics
11681:Introduction to Probability
11415:Introduction to Probability
11315:Algebra of random variables
11286:
11114:
10868:moment generating functions
10703:âthey are not closed under
7506:{\displaystyle (\Omega ,P)}
6823:, the relation between the
5364:. Moreover, when the space
4600:is a random variable whose
4157:falling in any subinterval
2725:
2298:and sometimes also using a
1424:natural language processing
10:
14300:
13699:Multivariate distributions
12119:Average absolute deviation
11522:(2nd ed.). New York:
11519:The Practice of Statistics
11320:Event (probability theory)
11210:
7607:continuous random variable
7315:In the measure-theoretic,
7070:{\displaystyle y=g(x_{i})}
5720:
4533:. Samples of any desired
4200:{\displaystyle \subseteq }
3847:everywhere. There are no "
3835:Continuous random variable
2666:correlation and dependence
2079:{\displaystyle P(X=2)\,\!}
2049:which is often written as
1260:of a random variable, its
1195:continuous random variable
1141:and its distribution is a
1133:is finitely or infinitely
805:to be a sample space of a
14213:
14167:
14104:
14057:
14020:
14016:
14003:
13975:
13957:
13924:
13915:
13873:
13820:
13781:
13730:
13721:
13687:Structural equation model
13642:
13599:
13595:
13570:
13529:
13495:
13449:
13416:
13378:
13345:
13341:
13317:
13257:
13166:
13085:
13049:
13040:
13023:Score/Lagrange multiplier
13008:
12961:
12906:
12832:
12823:
12633:
12629:
12616:
12575:
12549:
12501:
12456:
12438:Sample size determination
12403:
12399:
12386:
12290:
12245:
12219:
12201:
12157:
12109:
12029:
12020:
12016:
12003:
11985:
11833:Kallenberg, Olav (2001).
11738:Fristedt & Gray (1996
9992:, because every value of
9158:, because every value of
7361:Borel measurable function
6195:Borel measurable function
6153:, and has the value 1 if
5985:{\displaystyle \{f_{i}\}}
4823:is a measurable function
4582:randomly-generated number
3974:More formally, given any
3553:probability mass function
3378:probability mass function
3184:{\displaystyle \{a_{n}\}}
1730:{\displaystyle N\times N}
1147:probability mass function
629:discrete random variables
598:probability measure space
14182:Environmental statistics
13704:Elliptical distributions
13497:Generalized linear model
13426:Simple linear regression
13196:HodgesâLehmann estimator
12653:Probability distribution
12562:Stochastic approximation
12124:Coefficient of variation
11938:Basic Probability Topics
11935:Zukerman, Moshe (2014),
11912:Zukerman, Moshe (2014),
11839:(2nd ed.). Berlin:
11806:(4th ed.). Berlin:
11712:Steigerwald, Douglas G.
10731:Equality in distribution
9828:{\displaystyle Y=X^{2}.}
9632:chi-squared distribution
8994:{\displaystyle Y=X^{2}.}
8840:exponential distribution
7290:inverse function theorem
7081:can be generalized with
6495:increasing or decreasing
6442:{\displaystyle h=g^{-1}}
4535:probability distribution
2731:Discrete random variable
2722:for fuller development.
2565:RadonâNikodym derivative
2147:probability distribution
2115:{\displaystyle p_{X}(2)}
1139:discrete random variable
509:{\displaystyle \{-1,1\}}
285:Law of total probability
280:Conditional independence
169:Exponential distribution
154:Probability distribution
13842:Cross-correlation (XCF)
13450:Non-standard predictors
12884:LehmannâScheffĂŠ theorem
12557:Adaptive clinical trial
11867:(9th ed.). Tokyo:
11779:Probability and Measure
11649:Probability and Measure
11360:Random number generator
7877:{\displaystyle y\geq 0}
7635:{\displaystyle Y=X^{2}}
7446:{\displaystyle \Omega }
7352:{\displaystyle \Omega }
6044:of the random variable
6040:fully characterise the
5362:-valued random variable
5080:{\displaystyle \Omega }
4821:-valued random variable
3151:{\displaystyle x\geq t}
2903:{\textstyle b_{n}>0}
2689:{\displaystyle \Omega }
2657:{\displaystyle \Omega }
2386:{\displaystyle \Omega }
1475:{\displaystyle \Omega }
1405:-valued random variable
889:{\displaystyle X,Y,Z,T}
798:{\displaystyle \Omega }
751:{\displaystyle \Omega }
423:{\displaystyle \{H,T\}}
390:is the set of possible
264:Conditional probability
14284:Statistical randomness
14238:Mathematics portal
14059:Engineering statistics
13967:NelsonâAalen estimator
13544:Analysis of covariance
13431:Ordinary least squares
13355:Pearson product-moment
12759:Statistical functional
12670:Empirical distribution
12503:Controlled experiments
12232:Frequency distribution
12010:Descriptive statistics
11761:. Boston: Birkhäuser.
11273:
11253:
11186:
11094:
10987:
10936:
10857:
10780:
10664:
10440:
10348:and its derivative is
10339:
10264:
10026:
10006:
9979:
9829:
9790:
9660:
9621:
9473:
9395:and its derivative is
9386:
9325:
9192:
9172:
9145:
8995:
8956:
8864:
8825:
8544:
8518:
8346:
8284:
8255:
8151:
8123:
8008:
7878:
7849:
7786:
7738:
7737:{\displaystyle y<0}
7709:
7636:
7599:
7571:
7551:
7507:
7471:
7447:
7427:
7426:{\displaystyle Y=g(X)}
7392:
7353:
7333:
7306:
7282:
7222:
7071:
7029:
7002:
6982:
6959:
6841:
6813:
6790:
6483:
6463:
6443:
6407:
6384:
6309:
6285:
6284:{\displaystyle Y=g(X)}
6250:
6226:
6185:A new random variable
6167:
6143:
6097:
6096:{\displaystyle f(X)=X}
6058:
6034:
5986:
5953:
5926:
5906:
5880:of a random variable.
5866:
5846:
5805:
5762:, and also called the
5756:
5711:
5624:
5566:
5488:
5454:
5400:
5378:
5356:
5335:
5299:
5279:
5243:
5212:
5189:
5169:
5145:
5125:
5105:
5081:
5058:
5028:
4956:
4907:
4879:
4849:
4815:
4774:
4734:
4574:
4550:
4527:
4492:
4370:
4316:
4201:
4151:
4116:
4047:
3916:
3826:
3652:
3556:
3526:
3400:
3360:
3255:
3235:
3185:
3152:
3126:
3084:
3083:{\displaystyle x<t}
3058:
3016:
2947:
2904:
2871:
2808:
2712:
2690:
2658:
2630:
2608:
2588:
2557:
2503:
2483:
2463:
2436:
2414:
2387:
2367:
2347:
2323:
2289:
2223:
2203:
2183:
2163:
2139:
2116:
2080:
2043:
1994:
1974:
1932:
1900:Distribution functions
1885:
1839:
1810:
1790:
1761:
1731:
1705:
1678:
1658:
1635:
1585:
1535:
1476:
1452:
1426:, and other fields in
1399:
1377:
1308:
1288:
1246:
1187:
1163:
1127:
1092:
1060:
1031:
939:
913:
890:
844:
799:
779:
752:
732:
697:
612:, which is called the
569:
550:
530:
510:
464:
444:
424:
206:Continuous or discrete
159:Bernoulli distribution
14154:Population statistics
14096:System identification
13830:Autocorrelation (ACF)
13758:Exponential smoothing
13672:Discriminant analysis
13667:Canonical correlation
13531:Partition of variance
13393:Regression validation
13237:(JonckheereâTerpstra)
13136:Likelihood-ratio test
12825:Frequentist inference
12737:Locationâscale family
12658:Sampling distribution
12623:Statistical inference
12590:Cross-sectional study
12577:Observational studies
12536:Randomized experiment
12365:Stem-and-leaf display
12167:Central limit theorem
11626:. Wiley. p. 67.
11589:10.1007/1-84628-168-7
11274:
11254:
11252:{\displaystyle X_{n}}
11227:central limit theorem
11187:
11095:
10988:
10937:
10886:Two random variables
10858:
10781:
10745:equal in distribution
10665:
10441:
10340:
10265:
10027:
10007:
9980:
9830:
9791:
9661:
9622:
9474:
9387:
9326:
9193:
9173:
9146:
8996:
8957:
8865:
8826:
8545:
8519:
8347:
8285:
8256:
8152:
8124:
8009:
7879:
7850:
7787:
7739:
7710:
7637:
7600:
7572:
7552:
7508:
7472:
7448:
7428:
7393:
7354:
7334:
7307:
7283:
7223:
7072:
7030:
7028:{\displaystyle x_{i}}
7003:
6983:
6960:
6847:, in order to obtain
6842:
6814:
6791:
6484:
6464:
6444:
6413:is invertible (i.e.,
6408:
6385:
6310:
6286:
6251:
6232:to the outcomes of a
6227:
6168:
6144:
6111:that can take on the
6098:
6059:
6035:
5987:
5954:
5927:
5907:
5867:
5847:
5806:
5757:
5712:
5625:
5567:
5489:
5455:
5401:
5379:
5357:
5336:
5300:
5280:
5244:
5213:
5190:
5170:
5146:
5126:
5106:
5082:
5059:
5029:
4957:
4908:
4880:
4850:
4816:
4775:
4735:
4670:BanachâTarski paradox
4610:everywhere-continuous
4598:mixed random variable
4575:
4551:
4528:
4493:
4371:
4328:of probability. The
4317:
4202:
4152:
4150:{\displaystyle X_{I}}
4117:
4048:
3940:absolutely continuous
3917:
3827:
3658:and (if the dice are
3653:
3542:
3527:
3401:
3399:{\displaystyle f_{Y}}
3361:
3256:
3236:
3186:
3153:
3127:
3085:
3059:
3017:
2948:
2905:
2872:
2809:
2713:
2691:
2659:
2631:
2609:
2589:
2587:{\displaystyle p_{X}}
2558:
2504:
2484:
2464:
2462:{\displaystyle p_{X}}
2437:
2415:
2413:{\displaystyle p_{X}}
2388:
2368:
2348:
2324:
2322:{\displaystyle f_{X}}
2290:
2224:
2204:
2184:
2164:
2140:
2117:
2081:
2044:
1995:
1975:
1933:
1904:If a random variable
1886:
1840:
1811:
1791:
1762:
1732:
1706:
1679:
1659:
1636:
1586:
1536:
1477:
1453:
1400:
1378:
1309:
1289:
1268:of its distribution.
1247:
1199:absolutely continuous
1188:
1164:
1128:
1093:
1061:
1032:
940:
914:
899:The probability that
891:
845:
800:
780:
758:as a set of possible
753:
733:
698:
567:
551:
531:
511:
465:
445:
425:
164:Binomial distribution
14077:Probabilistic design
13662:Principal components
13505:Exponential families
13457:Nonlinear regression
13436:General linear model
13398:Mixed effects models
13388:Errors and residuals
13365:Confounding variable
13267:Bayesian probability
13245:Van der Waerden test
13235:Ordered alternative
13000:Multiple comparisons
12879:RaoâBlackwellization
12842:Estimating equations
12798:Statistical distance
12516:Factorial experiment
12049:Arithmetic-Geometric
11859:Papoulis, Athanasios
11263:
11236:
11223:law of large numbers
11138:
11003:
10953:
10905:
10882:Almost sure equality
10793:
10751:
10456:
10355:
10280:
10039:
10016:
9996:
9842:
9803:
9677:
9650:
9489:
9402:
9341:
9205:
9182:
9162:
9008:
8969:
8881:
8854:
8557:
8531:
8359:
8294:
8268:
8164:
8141:
8024:
7891:
7862:
7799:
7748:
7722:
7649:
7613:
7589:
7561:
7517:
7485:
7461:
7437:
7402:
7366:
7343:
7323:
7296:
7235:
7088:
7039:
7012:
6992:
6972:
6854:
6831:
6803:
6504:
6473:
6453:
6417:
6397:
6322:
6299:
6260:
6240:
6200:
6157:
6119:
6072:
6048:
5996:
5963:
5959:, find a collection
5943:
5916:
5884:
5856:
5815:
5774:
5734:
5634:
5579:
5501:
5464:
5422:
5388:
5368:
5346:
5309:
5289:
5256:
5229:
5202:
5179:
5155:
5135:
5115:
5091:
5071:
5038:
4966:
4917:
4893:
4859:
4827:
4789:
4748:
4702:
4564:
4540:
4505:
4388:
4336:
4239:
4161:
4134:
4057:
4053:, a random variable
3981:
3936:probability measures
3867:
3675:
3581:
3410:
3383:
3265:
3245:
3203:
3162:
3136:
3094:
3068:
3026:
2957:
2914:
2881:
2829:
2744:
2700:
2680:
2648:
2618:
2607:{\displaystyle \mu }
2598:
2571:
2513:
2493:
2473:
2446:
2424:
2397:
2377:
2357:
2337:
2306:
2239:
2213:
2193:
2173:
2153:
2129:
2090:
2053:
2004:
1984:
1942:
1908:
1857:
1838:{\displaystyle F(x)}
1820:
1800:
1789:{\displaystyle F(x)}
1771:
1751:
1741:of the random graph.
1715:
1695:
1668:
1648:
1595:
1545:
1495:
1466:
1442:
1428:discrete mathematics
1389:
1367:
1298:
1278:
1228:
1177:
1153:
1117:
1074:
1050:
952:
923:
903:
862:
812:
789:
769:
742:
738:from a sample space
710:
687:
641:stochastic processes
540:
520:
485:
454:
434:
402:
290:Law of large numbers
259:Marginal probability
184:Poisson distribution
33:Part of a series on
14149:Official statistics
14072:Methods engineering
13753:Seasonal adjustment
13521:Poisson regressions
13441:Bayesian regression
13380:Regression analysis
13360:Partial correlation
13332:Regression analysis
12931:Prediction interval
12926:Likelihood interval
12916:Confidence interval
12908:Interval estimation
12869:Unbiased estimators
12687:Model specification
12567:Up-and-down designs
12255:Partial correlation
12211:Index of dispersion
12129:Interquartile range
11782:. New York: Wiley.
11335:Observable variable
10705:linear combinations
10387:
10312:
10232:
10192:
10134:
10094:
9947:
9907:
9670:, whose density is
9668:normal distribution
9113:
9073:
8874:, whose density is
7479:Lebesgue measurable
7288:, according to the
7268:
7193:
7153:
5305:. In such case the
4692:of such intervals.
4380:of its interval of
1360:random variable of
705:measurable function
618:probability measure
610:pushforward measure
594:measurable function
368:stochastic variable
249:Complementary event
191:Probability measure
179:Pareto distribution
174:Normal distribution
14169:Spatial statistics
14049:Medical statistics
13949:First hitting time
13903:Whittle likelihood
13554:Degrees of freedom
13549:Multivariate ANOVA
13482:Heteroscedasticity
13294:Bayesian estimator
13259:Bayesian inference
13108:KolmogorovâSmirnov
12993:Randomization test
12963:Testing hypotheses
12936:Tolerance interval
12847:Maximum likelihood
12742:Exponential family
12675:Density estimation
12635:Statistical theory
12595:Natural experiment
12541:Scientific control
12458:Survey methodology
12144:Standard deviation
11552:"Random Variables"
11497:www.mathsisfun.com
11493:"Random Variables"
11380:Stochastic process
11340:Random compact set
11301:Mathematics portal
11269:
11249:
11182:
11174:
11105:essential supremum
11090:
11049:
10983:
10932:
10853:
10776:
10709:convex combination
10660:
10436:
10364:
10335:
10289:
10260:
10215:
10175:
10117:
10077:
10022:
10002:
9975:
9930:
9890:
9876:
9825:
9786:
9656:
9617:
9469:
9382:
9321:
9188:
9168:
9141:
9096:
9056:
9042:
8991:
8952:
8860:
8821:
8819:
8543:{\displaystyle X,}
8540:
8514:
8342:
8280:
8251:
8147:
8119:
8107:
8004:
7874:
7845:
7833:
7782:
7734:
7705:
7632:
7605:be a real-valued,
7595:
7567:
7547:
7503:
7467:
7455:is also measurable
7443:
7423:
7388:
7349:
7329:
7317:axiomatic approach
7312:to be increasing.
7302:
7278:
7251:
7218:
7176:
7136:
7122:
7067:
7025:
6998:
6978:
6955:
6837:
6809:
6786:
6781:
6479:
6459:
6439:
6403:
6380:
6305:
6281:
6246:
6222:
6189:can be defined by
6163:
6139:
6093:
6054:
6030:
5982:
5949:
5937:problem of moments
5922:
5902:
5878:standard deviation
5862:
5842:
5801:
5752:
5707:
5620:
5562:
5484:
5450:
5396:
5374:
5352:
5331:
5295:
5275:
5239:
5208:
5185:
5165:
5141:
5121:
5101:
5077:
5054:
5024:
4952:
4903:
4875:
4845:
4811:
4770:
4730:
4682:countably infinite
4570:
4546:
4523:
4488:
4483:
4443:
4378:indicator function
4366:
4312:
4197:
4147:
4124:continuous uniform
4112:
4043:
3912:
3822:
3648:
3557:
3522:
3517:
3491:
3453:
3396:
3356:
3351:
3251:
3231:
3181:
3148:
3122:
3080:
3054:
3012:
2975:
2943:
2926:
2900:
2867:
2804:
2720:quantile functions
2708:
2686:
2674:joint distribution
2654:
2626:
2604:
2584:
2553:
2499:
2479:
2459:
2432:
2410:
2383:
2363:
2343:
2319:
2285:
2219:
2199:
2179:
2159:
2135:
2112:
2076:
2039:
1990:
1970:
1928:
1881:
1847:stochastic process
1835:
1806:
1786:
1757:
1727:
1701:
1674:
1654:
1631:
1581:
1531:
1472:
1448:
1395:
1373:
1320:categorical values
1304:
1284:
1242:
1183:
1159:
1123:
1088:
1056:
1027:
935:
909:
886:
840:
807:probability triple
795:
775:
748:
728:
693:
570:
546:
536:mapped to -1 and
526:
506:
460:
440:
420:
300:Boole's inequality
236:Stochastic process
125:Mutual exclusivity
42:Probability theory
14271:
14270:
14209:
14208:
14205:
14204:
14144:National accounts
14114:Actuarial science
14106:Social statistics
13999:
13998:
13995:
13994:
13991:
13990:
13926:Survival function
13911:
13910:
13773:Granger causality
13614:Contingency table
13589:Survival analysis
13566:
13565:
13562:
13561:
13418:Linear regression
13313:
13312:
13309:
13308:
13284:Credible interval
13253:
13252:
13036:
13035:
12852:Method of moments
12721:Parametric family
12682:Statistical model
12612:
12611:
12608:
12607:
12526:Random assignment
12448:Statistical power
12382:
12381:
12378:
12377:
12227:Contingency table
12197:
12196:
12064:Generalized/power
11895:"Random variable"
11598:978-1-85233-896-1
11556:www.stat.yale.edu
11533:978-0-7167-4773-4
11451:978-1-108-47004-9
11272:{\displaystyle X}
11173:
11040:
10925:
10923:
10876:Laplace transform
10845:
10770:
10711:, thus forming a
10679:degree of freedom
10610:
10544:
10523:
10520:
10504:
10503:
10431:
10428:
10406:
10333:
10251:
10153:
10025:{\displaystyle X}
10005:{\displaystyle Y}
9966:
9867:
9725:
9724:
9659:{\displaystyle X}
9636:degree of freedom
9591:
9590:
9570:
9567:
9530:
9529:
9464:
9461:
9442:
9380:
9312:
9191:{\displaystyle X}
9171:{\displaystyle Y}
9132:
9033:
8919:
8918:
8863:{\displaystyle X}
8780:
8720:
8249:
8150:{\displaystyle X}
8106:
8096:
8067:
7996:
7980:
7955:
7832:
7598:{\displaystyle X}
7570:{\displaystyle Y}
7470:{\displaystyle g}
7332:{\displaystyle X}
7305:{\displaystyle g}
7212:
7113:
7001:{\displaystyle y}
6981:{\displaystyle g}
6946:
6840:{\displaystyle y}
6821:differentiability
6812:{\displaystyle g}
6774:
6750:
6663:
6639:
6482:{\displaystyle g}
6462:{\displaystyle h}
6406:{\displaystyle g}
6308:{\displaystyle Y}
6249:{\displaystyle X}
6166:{\displaystyle X}
6134:
6057:{\displaystyle X}
5952:{\displaystyle X}
5925:{\displaystyle X}
5865:{\displaystyle X}
5384:is the real line
5377:{\displaystyle E}
5355:{\displaystyle E}
5298:{\displaystyle E}
5220:topological space
5211:{\displaystyle E}
5188:{\displaystyle E}
5144:{\displaystyle E}
5124:{\displaystyle P}
4742:probability space
4656:The most formal,
4558:quantile function
4476:
4438:
4310:
3928:arbitrarily small
3745:
3737:
3501:
3490:
3463:
3452:
3368:If the coin is a
3344:
3333:
3313:
3302:
3254:{\displaystyle Y}
3226:
3218:
2966:
2917:
2502:{\displaystyle X}
2489:" or the "law of
2482:{\displaystyle X}
2366:{\displaystyle P}
2346:{\displaystyle X}
2331:measure-theoretic
2222:{\displaystyle X}
2202:{\displaystyle X}
2182:{\displaystyle X}
2162:{\displaystyle X}
2138:{\displaystyle X}
1993:{\displaystyle X}
1809:{\displaystyle x}
1760:{\displaystyle F}
1704:{\displaystyle N}
1677:{\displaystyle N}
1657:{\displaystyle N}
1624:
1618:
1612:
1606:
1574:
1568:
1562:
1556:
1524:
1518:
1512:
1506:
1451:{\displaystyle E}
1398:{\displaystyle E}
1376:{\displaystyle E}
1314:, such as random
1307:{\displaystyle E}
1287:{\displaystyle E}
1186:{\displaystyle X}
1162:{\displaystyle X}
1126:{\displaystyle X}
1059:{\displaystyle X}
912:{\displaystyle X}
778:{\displaystyle E}
696:{\displaystyle X}
670:Pafnuty Chebyshev
578:measurement error
549:{\displaystyle T}
529:{\displaystyle H}
463:{\displaystyle T}
443:{\displaystyle H}
364:aleatory variable
352:
351:
254:Joint probability
201:Bernoulli process
100:Probability space
16:(Redirected from
14291:
14259:
14258:
14247:
14246:
14236:
14235:
14221:
14220:
14124:Crime statistics
14018:
14017:
14005:
14004:
13922:
13921:
13888:Fourier analysis
13875:Frequency domain
13855:
13802:
13768:Structural break
13728:
13727:
13677:Cluster analysis
13624:Log-linear model
13597:
13596:
13572:
13571:
13513:
13487:Homoscedasticity
13343:
13342:
13319:
13318:
13238:
13230:
13222:
13221:(KruskalâWallis)
13206:
13191:
13146:Cross validation
13131:
13113:AndersonâDarling
13060:
13047:
13046:
13018:Likelihood-ratio
13010:Parametric tests
12988:Permutation test
12971:1- & 2-tails
12862:Minimum distance
12834:Point estimation
12830:
12829:
12781:Optimal decision
12732:
12631:
12630:
12618:
12617:
12600:Quasi-experiment
12550:Adaptive designs
12401:
12400:
12388:
12387:
12265:Rank correlation
12027:
12026:
12018:
12017:
12005:
12004:
11972:
11965:
11958:
11949:
11948:
11944:
11943:
11931:
11930:
11920:
11908:
11882:
11854:
11829:
11798:Kallenberg, Olav
11793:
11772:
11741:
11735:
11729:
11728:
11726:
11724:
11718:
11709:
11703:
11702:
11676:
11663:
11662:
11644:
11638:
11637:
11617:
11611:
11610:
11572:
11566:
11565:
11563:
11562:
11548:
11542:
11541:
11536:. Archived from
11513:
11507:
11506:
11504:
11503:
11489:
11483:
11482:
11470:
11464:
11463:
11435:
11429:
11428:
11410:
11397:Inline citations
11303:
11298:
11297:
11278:
11276:
11275:
11270:
11258:
11256:
11255:
11250:
11248:
11247:
11191:
11189:
11188:
11183:
11175:
11171:
11107:in the sense of
11099:
11097:
11096:
11091:
11086:
11054:
11048:
11015:
11014:
10992:
10990:
10989:
10984:
10941:
10939:
10938:
10933:
10927:
10926:
10924:
10921:
10919:
10914:
10862:
10860:
10859:
10854:
10846:
10843:
10785:
10783:
10782:
10777:
10772:
10771:
10769:
10764:
10759:
10669:
10667:
10666:
10661:
10653:
10652:
10648:
10647:
10632:
10627:
10626:
10611:
10606:
10587:
10586:
10582:
10581:
10566:
10561:
10560:
10545:
10540:
10524:
10522:
10521:
10516:
10507:
10505:
10502:
10501:
10486:
10482:
10468:
10467:
10445:
10443:
10442:
10437:
10432:
10430:
10429:
10424:
10415:
10407:
10405:
10397:
10386:
10378:
10359:
10344:
10342:
10341:
10336:
10334:
10329:
10311:
10303:
10269:
10267:
10266:
10261:
10256:
10252:
10250:
10242:
10231:
10223:
10210:
10191:
10183:
10171:
10170:
10158:
10154:
10152:
10144:
10133:
10125:
10112:
10093:
10085:
10073:
10072:
10051:
10050:
10031:
10029:
10028:
10023:
10011:
10009:
10008:
10003:
9984:
9982:
9981:
9976:
9971:
9967:
9965:
9957:
9946:
9938:
9925:
9906:
9898:
9886:
9885:
9875:
9854:
9853:
9834:
9832:
9831:
9826:
9821:
9820:
9795:
9793:
9792:
9787:
9782:
9781:
9777:
9776:
9761:
9756:
9755:
9726:
9723:
9722:
9707:
9703:
9689:
9688:
9665:
9663:
9662:
9657:
9626:
9624:
9623:
9618:
9613:
9612:
9608:
9592:
9580:
9576:
9571:
9569:
9568:
9563:
9554:
9552:
9551:
9547:
9531:
9522:
9518:
9501:
9500:
9478:
9476:
9475:
9470:
9465:
9463:
9462:
9457:
9448:
9443:
9441:
9433:
9423:
9422:
9406:
9391:
9389:
9388:
9383:
9381:
9376:
9362:
9361:
9330:
9328:
9327:
9322:
9317:
9313:
9311:
9303:
9293:
9292:
9276:
9258:
9257:
9242:
9241:
9217:
9216:
9197:
9195:
9194:
9189:
9177:
9175:
9174:
9169:
9150:
9148:
9147:
9142:
9137:
9133:
9131:
9123:
9112:
9104:
9091:
9072:
9064:
9052:
9051:
9041:
9020:
9019:
9000:
8998:
8997:
8992:
8987:
8986:
8961:
8959:
8958:
8953:
8948:
8947:
8943:
8938:
8937:
8920:
8911:
8907:
8893:
8892:
8869:
8867:
8866:
8861:
8830:
8828:
8827:
8822:
8820:
8813:
8812:
8785:
8781:
8779:
8778:
8777:
8762:
8761:
8739:
8725:
8721:
8719:
8718:
8717:
8708:
8707:
8697:
8696:
8660:
8646:
8630:
8629:
8605:
8604:
8573:
8572:
8549:
8547:
8546:
8541:
8523:
8521:
8520:
8515:
8497:
8496:
8484:
8443:
8442:
8421:
8371:
8370:
8351:
8349:
8348:
8343:
8335:
8334:
8313:
8289:
8287:
8286:
8281:
8260:
8258:
8257:
8252:
8250:
8248:
8247:
8246:
8237:
8236:
8211:
8176:
8175:
8156:
8154:
8153:
8148:
8128:
8126:
8125:
8120:
8108:
8104:
8097:
8092:
8084:
8083:
8068:
8063:
8058:
8057:
8036:
8035:
8013:
8011:
8010:
8005:
7997:
7992:
7981:
7976:
7956:
7951:
7946:
7938:
7912:
7911:
7883:
7881:
7880:
7875:
7854:
7852:
7851:
7846:
7834:
7830:
7811:
7810:
7791:
7789:
7788:
7783:
7766:
7765:
7743:
7741:
7740:
7735:
7714:
7712:
7711:
7706:
7692:
7691:
7661:
7660:
7641:
7639:
7638:
7633:
7631:
7630:
7604:
7602:
7601:
7596:
7576:
7574:
7573:
7568:
7556:
7554:
7553:
7548:
7543:
7542:
7527:
7512:
7510:
7509:
7504:
7476:
7474:
7473:
7468:
7452:
7450:
7449:
7444:
7432:
7430:
7429:
7424:
7397:
7395:
7394:
7389:
7387:
7379:
7358:
7356:
7355:
7350:
7338:
7336:
7335:
7330:
7311:
7309:
7308:
7303:
7287:
7285:
7284:
7279:
7267:
7259:
7247:
7246:
7227:
7225:
7224:
7219:
7217:
7213:
7211:
7203:
7192:
7184:
7171:
7152:
7144:
7132:
7131:
7121:
7100:
7099:
7076:
7074:
7073:
7068:
7063:
7062:
7034:
7032:
7031:
7026:
7024:
7023:
7007:
7005:
7004:
6999:
6987:
6985:
6984:
6979:
6964:
6962:
6961:
6956:
6951:
6947:
6945:
6937:
6920:
6914:
6913:
6895:
6894:
6888:
6887:
6866:
6865:
6846:
6844:
6843:
6838:
6819:, assuming also
6818:
6816:
6815:
6810:
6795:
6793:
6792:
6787:
6785:
6784:
6775:
6773: decreasing
6772:
6770:
6769:
6751:
6748:
6723:
6722:
6671:
6664:
6662: increasing
6661:
6659:
6658:
6640:
6637:
6612:
6611:
6516:
6515:
6493:) and is either
6491:inverse function
6488:
6486:
6485:
6480:
6468:
6466:
6465:
6460:
6448:
6446:
6445:
6440:
6438:
6437:
6412:
6410:
6409:
6404:
6389:
6387:
6386:
6381:
6334:
6333:
6314:
6312:
6311:
6306:
6290:
6288:
6287:
6282:
6255:
6253:
6252:
6247:
6236:random variable
6231:
6229:
6228:
6223:
6221:
6213:
6172:
6170:
6169:
6164:
6148:
6146:
6145:
6142:{\displaystyle }
6140:
6135:
6132:
6107:random variable
6102:
6100:
6099:
6094:
6063:
6061:
6060:
6055:
6039:
6037:
6036:
6031:
6017:
6016:
5991:
5989:
5988:
5983:
5978:
5977:
5958:
5956:
5955:
5950:
5931:
5929:
5928:
5923:
5911:
5909:
5908:
5903:
5871:
5869:
5868:
5863:
5851:
5849:
5848:
5843:
5811:is not equal to
5810:
5808:
5807:
5802:
5761:
5759:
5758:
5753:
5716:
5714:
5713:
5708:
5682:
5681:
5629:
5627:
5626:
5621:
5616:
5571:
5569:
5568:
5563:
5558:
5543:
5542:
5493:
5491:
5490:
5485:
5483:
5459:
5457:
5456:
5451:
5440:
5439:
5405:
5403:
5402:
5397:
5395:
5383:
5381:
5380:
5375:
5361:
5359:
5358:
5353:
5340:
5338:
5337:
5332:
5327:
5326:
5304:
5302:
5301:
5296:
5284:
5282:
5281:
5276:
5265:
5264:
5248:
5246:
5245:
5240:
5238:
5237:
5217:
5215:
5214:
5209:
5194:
5192:
5191:
5186:
5174:
5172:
5171:
5166:
5164:
5163:
5150:
5148:
5147:
5142:
5130:
5128:
5127:
5122:
5110:
5108:
5107:
5102:
5100:
5099:
5086:
5084:
5083:
5078:
5063:
5061:
5060:
5055:
5053:
5052:
5033:
5031:
5030:
5025:
4981:
4980:
4961:
4959:
4958:
4953:
4951:
4950:
4932:
4931:
4912:
4910:
4909:
4904:
4902:
4901:
4884:
4882:
4881:
4876:
4874:
4873:
4854:
4852:
4851:
4846:
4820:
4818:
4817:
4812:
4807:
4806:
4782:measurable space
4779:
4777:
4776:
4771:
4766:
4765:
4739:
4737:
4736:
4731:
4720:
4719:
4640:
4639:
4635:
4625:= â1; otherwise
4615:
4579:
4577:
4576:
4571:
4555:
4553:
4552:
4547:
4532:
4530:
4529:
4526:{\displaystyle }
4524:
4497:
4495:
4494:
4489:
4487:
4486:
4477:
4474:
4439:
4437:
4423:
4400:
4399:
4376:is given by the
4375:
4373:
4372:
4367:
4321:
4319:
4318:
4313:
4311:
4309:
4298:
4287:
4282:
4278:
4259:
4258:
4232:
4206:
4204:
4203:
4198:
4156:
4154:
4153:
4148:
4146:
4145:
4121:
4119:
4118:
4113:
4069:
4068:
4052:
4050:
4049:
4044:
4021:
3959:
3958:
3954:
3921:
3919:
3918:
3913:
3883:
3831:
3829:
3828:
3823:
3746:
3743:
3738:
3733:
3701:
3687:
3686:
3657:
3655:
3654:
3649:
3647:
3646:
3634:
3633:
3615:
3614:
3602:
3601:
3531:
3529:
3528:
3523:
3521:
3520:
3502:
3499:
3492:
3483:
3464:
3461:
3454:
3445:
3422:
3421:
3405:
3403:
3402:
3397:
3395:
3394:
3365:
3363:
3362:
3357:
3355:
3354:
3345:
3342:
3334:
3331:
3314:
3311:
3303:
3300:
3260:
3258:
3257:
3252:
3240:
3238:
3237:
3232:
3227:
3224:
3219:
3216:
3190:
3188:
3187:
3182:
3177:
3176:
3157:
3155:
3154:
3149:
3131:
3129:
3128:
3123:
3106:
3105:
3089:
3087:
3086:
3081:
3063:
3061:
3060:
3055:
3038:
3037:
3021:
3019:
3018:
3013:
3002:
3001:
3000:
2999:
2985:
2984:
2974:
2952:
2950:
2949:
2944:
2936:
2935:
2925:
2909:
2907:
2906:
2901:
2893:
2892:
2876:
2874:
2873:
2868:
2863:
2862:
2844:
2843:
2813:
2811:
2810:
2805:
2717:
2715:
2714:
2709:
2707:
2695:
2693:
2692:
2687:
2663:
2661:
2660:
2655:
2642:counting measure
2638:Lebesgue measure
2635:
2633:
2632:
2627:
2625:
2613:
2611:
2610:
2605:
2593:
2591:
2590:
2585:
2583:
2582:
2562:
2560:
2559:
2554:
2546:
2541:
2540:
2525:
2524:
2509:". The density
2508:
2506:
2505:
2500:
2488:
2486:
2485:
2480:
2468:
2466:
2465:
2460:
2458:
2457:
2441:
2439:
2438:
2433:
2431:
2419:
2417:
2416:
2411:
2409:
2408:
2392:
2390:
2389:
2384:
2372:
2370:
2369:
2364:
2352:
2350:
2349:
2344:
2328:
2326:
2325:
2320:
2318:
2317:
2294:
2292:
2291:
2286:
2251:
2250:
2228:
2226:
2225:
2220:
2208:
2206:
2205:
2200:
2188:
2186:
2185:
2180:
2168:
2166:
2165:
2160:
2144:
2142:
2141:
2136:
2121:
2119:
2118:
2113:
2102:
2101:
2085:
2083:
2082:
2077:
2048:
2046:
2045:
2040:
1999:
1997:
1996:
1991:
1979:
1977:
1976:
1971:
1960:
1959:
1937:
1935:
1934:
1929:
1927:
1890:
1888:
1887:
1882:
1844:
1842:
1841:
1836:
1815:
1813:
1812:
1807:
1795:
1793:
1792:
1787:
1766:
1764:
1763:
1758:
1739:adjacency matrix
1736:
1734:
1733:
1728:
1710:
1708:
1707:
1702:
1683:
1681:
1680:
1675:
1663:
1661:
1660:
1655:
1640:
1638:
1637:
1632:
1622:
1616:
1610:
1604:
1590:
1588:
1587:
1582:
1572:
1566:
1560:
1554:
1540:
1538:
1537:
1532:
1522:
1516:
1510:
1504:
1486:). For example:
1481:
1479:
1478:
1473:
1457:
1455:
1454:
1449:
1432:computer science
1420:machine learning
1404:
1402:
1401:
1396:
1382:
1380:
1379:
1374:
1313:
1311:
1310:
1305:
1293:
1291:
1290:
1285:
1273:measurable space
1251:
1249:
1248:
1243:
1241:
1192:
1190:
1189:
1184:
1168:
1166:
1165:
1160:
1132:
1130:
1129:
1124:
1097:
1095:
1094:
1089:
1087:
1065:
1063:
1062:
1057:
1036:
1034:
1033:
1028:
944:
942:
941:
936:
918:
916:
915:
910:
895:
893:
892:
887:
849:
847:
846:
841:
830:
829:
804:
802:
801:
796:
784:
782:
781:
776:
764:measurable space
757:
755:
754:
749:
737:
735:
734:
729:
702:
700:
699:
694:
649:random functions
645:random sequences
606:measurable space
555:
553:
552:
547:
535:
533:
532:
527:
515:
513:
512:
507:
479:measurable space
469:
467:
466:
461:
449:
447:
446:
441:
429:
427:
426:
421:
344:
337:
330:
120:Elementary event
52:
30:
29:
21:
18:Random variables
14299:
14298:
14294:
14293:
14292:
14290:
14289:
14288:
14274:
14273:
14272:
14267:
14230:
14201:
14163:
14100:
14086:quality control
14053:
14035:Clinical trials
14012:
13987:
13971:
13959:Hazard function
13953:
13907:
13869:
13853:
13816:
13812:BreuschâGodfrey
13800:
13777:
13717:
13692:Factor analysis
13638:
13619:Graphical model
13591:
13558:
13525:
13511:
13491:
13445:
13412:
13374:
13337:
13336:
13305:
13249:
13236:
13228:
13220:
13204:
13189:
13168:Rank statistics
13162:
13141:Model selection
13129:
13087:Goodness of fit
13081:
13058:
13032:
13004:
12957:
12902:
12891:Median unbiased
12819:
12730:
12663:Order statistic
12625:
12604:
12571:
12545:
12497:
12452:
12395:
12393:Data collection
12374:
12286:
12241:
12215:
12193:
12153:
12105:
12022:Continuous data
12012:
11999:
11981:
11976:
11941:
11918:
11893:
11890:
11885:
11879:
11851:
11841:Springer Verlag
11818:
11808:Akademie Verlag
11803:Random Measures
11790:
11769:
11750:
11745:
11744:
11736:
11732:
11722:
11720:
11716:
11710:
11706:
11691:
11677:
11666:
11659:
11645:
11641:
11634:
11618:
11614:
11599:
11573:
11569:
11560:
11558:
11550:
11549:
11545:
11534:
11514:
11510:
11501:
11499:
11491:
11490:
11486:
11471:
11467:
11452:
11436:
11432:
11425:
11411:
11404:
11399:
11394:
11389:
11350:Random function
11299:
11292:
11289:
11264:
11261:
11260:
11243:
11239:
11237:
11234:
11233:
11215:
11209:
11169:
11139:
11136:
11135:
11117:
11082:
11050:
11044:
11010:
11006:
11004:
11001:
11000:
10954:
10951:
10950:
10920:
10915:
10913:
10912:
10906:
10903:
10902:
10884:
10842:
10794:
10791:
10790:
10765:
10760:
10758:
10757:
10752:
10749:
10748:
10733:
10722:
10687:
10685:Some properties
10643:
10639:
10628:
10622:
10618:
10605:
10595:
10591:
10577:
10573:
10562:
10556:
10552:
10539:
10532:
10528:
10515:
10511:
10506:
10497:
10493:
10481:
10463:
10459:
10457:
10454:
10453:
10423:
10419:
10414:
10398:
10379:
10368:
10360:
10358:
10356:
10353:
10352:
10328:
10304:
10293:
10281:
10278:
10277:
10243:
10224:
10219:
10211:
10209:
10205:
10184:
10179:
10166:
10162:
10145:
10126:
10121:
10113:
10111:
10107:
10086:
10081:
10068:
10064:
10046:
10042:
10040:
10037:
10036:
10017:
10014:
10013:
9997:
9994:
9993:
9958:
9939:
9934:
9926:
9924:
9920:
9899:
9894:
9881:
9877:
9871:
9849:
9845:
9843:
9840:
9839:
9816:
9812:
9804:
9801:
9800:
9772:
9768:
9757:
9751:
9747:
9731:
9727:
9718:
9714:
9702:
9684:
9680:
9678:
9675:
9674:
9651:
9648:
9647:
9644:
9604:
9597:
9593:
9575:
9562:
9558:
9553:
9543:
9536:
9532:
9517:
9496:
9492:
9490:
9487:
9486:
9456:
9452:
9447:
9434:
9415:
9411:
9407:
9405:
9403:
9400:
9399:
9375:
9354:
9350:
9342:
9339:
9338:
9304:
9285:
9281:
9277:
9275:
9271:
9250:
9246:
9237:
9233:
9212:
9208:
9206:
9203:
9202:
9183:
9180:
9179:
9163:
9160:
9159:
9124:
9105:
9100:
9092:
9090:
9086:
9065:
9060:
9047:
9043:
9037:
9015:
9011:
9009:
9006:
9005:
8982:
8978:
8970:
8967:
8966:
8939:
8933:
8929:
8925:
8921:
8906:
8888:
8884:
8882:
8879:
8878:
8855:
8852:
8851:
8848:
8818:
8817:
8802:
8798:
8783:
8782:
8773:
8769:
8757:
8753:
8743:
8738:
8723:
8722:
8713:
8709:
8692:
8688:
8678:
8674:
8664:
8659:
8644:
8643:
8625:
8621:
8600:
8596:
8583:
8568:
8564:
8560:
8558:
8555:
8554:
8532:
8529:
8528:
8492:
8488:
8474:
8435:
8431:
8411:
8366:
8362:
8360:
8357:
8356:
8327:
8323:
8303:
8295:
8292:
8291:
8269:
8266:
8265:
8242:
8238:
8229:
8225:
8215:
8210:
8171:
8167:
8165:
8162:
8161:
8142:
8139:
8138:
8135:
8102:
8091:
8079:
8075:
8062:
8053:
8049:
8031:
8027:
8025:
8022:
8021:
7991:
7975:
7950:
7942:
7934:
7907:
7903:
7892:
7889:
7888:
7863:
7860:
7859:
7828:
7806:
7802:
7800:
7797:
7796:
7761:
7757:
7749:
7746:
7745:
7723:
7720:
7719:
7687:
7683:
7656:
7652:
7650:
7647:
7646:
7626:
7622:
7614:
7611:
7610:
7590:
7587:
7586:
7583:
7562:
7559:
7558:
7538:
7534:
7523:
7518:
7515:
7514:
7486:
7483:
7482:
7462:
7459:
7458:
7438:
7435:
7434:
7403:
7400:
7399:
7383:
7375:
7367:
7364:
7363:
7344:
7341:
7340:
7324:
7321:
7320:
7297:
7294:
7293:
7260:
7255:
7242:
7238:
7236:
7233:
7232:
7204:
7185:
7180:
7172:
7170:
7166:
7145:
7140:
7127:
7123:
7117:
7095:
7091:
7089:
7086:
7085:
7058:
7054:
7040:
7037:
7036:
7019:
7015:
7013:
7010:
7009:
6993:
6990:
6989:
6973:
6970:
6969:
6938:
6921:
6919:
6915:
6909:
6908:
6890:
6889:
6883:
6879:
6861:
6857:
6855:
6852:
6851:
6832:
6829:
6828:
6804:
6801:
6800:
6780:
6779:
6771:
6762:
6758:
6747:
6745:
6718:
6714:
6672:
6669:
6668:
6660:
6651:
6647:
6636:
6634:
6607:
6603:
6563:
6562:
6511:
6507:
6505:
6502:
6501:
6474:
6471:
6470:
6454:
6451:
6450:
6430:
6426:
6418:
6415:
6414:
6398:
6395:
6394:
6329:
6325:
6323:
6320:
6319:
6300:
6297:
6296:
6261:
6258:
6257:
6241:
6238:
6237:
6217:
6209:
6201:
6198:
6197:
6183:
6158:
6155:
6154:
6151:Iverson bracket
6131:
6120:
6117:
6116:
6073:
6070:
6069:
6049:
6046:
6045:
6012:
6008:
5997:
5994:
5993:
5973:
5969:
5964:
5961:
5960:
5944:
5941:
5940:
5917:
5914:
5913:
5885:
5882:
5881:
5857:
5854:
5853:
5816:
5813:
5812:
5775:
5772:
5771:
5735:
5732:
5731:
5723:
5674:
5670:
5635:
5632:
5631:
5612:
5580:
5577:
5576:
5554:
5538:
5537:
5502:
5499:
5498:
5479:
5465:
5462:
5461:
5435:
5434:
5423:
5420:
5419:
5416:
5408:random variable
5391:
5389:
5386:
5385:
5369:
5366:
5365:
5347:
5344:
5343:
5322:
5321:
5310:
5307:
5306:
5290:
5287:
5286:
5260:
5259:
5257:
5254:
5253:
5251:Borel Ď-algebra
5233:
5232:
5230:
5227:
5226:
5203:
5200:
5199:
5180:
5177:
5176:
5159:
5158:
5156:
5153:
5152:
5136:
5133:
5132:
5116:
5113:
5112:
5095:
5094:
5092:
5089:
5088:
5072:
5069:
5068:
5048:
5047:
5039:
5036:
5035:
4973:
4969:
4967:
4964:
4963:
4946:
4945:
4924:
4920:
4918:
4915:
4914:
4897:
4896:
4894:
4891:
4890:
4869:
4868:
4860:
4857:
4856:
4828:
4825:
4824:
4802:
4801:
4790:
4787:
4786:
4761:
4760:
4749:
4746:
4745:
4715:
4714:
4703:
4700:
4699:
4678:Borel Ď-algebra
4654:
4637:
4633:
4632:
4594:
4565:
4562:
4561:
4541:
4538:
4537:
4506:
4503:
4502:
4482:
4481:
4473:
4471:
4462:
4461:
4444:
4427:
4422:
4414:
4413:
4395:
4391:
4389:
4386:
4385:
4337:
4334:
4333:
4326:unitarity axiom
4299:
4288:
4286:
4254:
4250:
4249:
4245:
4240:
4237:
4236:
4216:
4162:
4159:
4158:
4141:
4137:
4135:
4132:
4131:
4064:
4060:
4058:
4055:
4054:
4017:
3982:
3979:
3978:
3956:
3952:
3951:
3879:
3868:
3865:
3864:
3837:
3744: for
3742:
3702:
3700:
3682:
3678:
3676:
3673:
3672:
3670:
3642:
3638:
3629:
3625:
3610:
3606:
3597:
3593:
3582:
3579:
3578:
3572:
3565:
3537:
3516:
3515:
3498:
3496:
3481:
3478:
3477:
3460:
3458:
3443:
3436:
3435:
3417:
3413:
3411:
3408:
3407:
3390:
3386:
3384:
3381:
3380:
3350:
3349:
3341:
3330:
3328:
3319:
3318:
3310:
3299:
3297:
3284:
3283:
3266:
3263:
3262:
3246:
3243:
3242:
3223:
3215:
3204:
3201:
3200:
3197:
3172:
3168:
3163:
3160:
3159:
3137:
3134:
3133:
3101:
3097:
3095:
3092:
3091:
3069:
3066:
3065:
3033:
3029:
3027:
3024:
3023:
2995:
2991:
2990:
2986:
2980:
2976:
2970:
2958:
2955:
2954:
2931:
2927:
2921:
2915:
2912:
2911:
2888:
2884:
2882:
2879:
2878:
2858:
2854:
2839:
2835:
2830:
2827:
2826:
2745:
2742:
2741:
2733:
2728:
2703:
2701:
2698:
2697:
2681:
2678:
2677:
2649:
2646:
2645:
2621:
2619:
2616:
2615:
2599:
2596:
2595:
2578:
2574:
2572:
2569:
2568:
2542:
2536:
2532:
2520:
2516:
2514:
2511:
2510:
2494:
2491:
2490:
2474:
2471:
2470:
2453:
2449:
2447:
2444:
2443:
2427:
2425:
2422:
2421:
2404:
2400:
2398:
2395:
2394:
2378:
2375:
2374:
2358:
2355:
2354:
2338:
2335:
2334:
2313:
2309:
2307:
2304:
2303:
2246:
2242:
2240:
2237:
2236:
2214:
2211:
2210:
2194:
2191:
2190:
2174:
2171:
2170:
2154:
2151:
2150:
2130:
2127:
2126:
2097:
2093:
2091:
2088:
2087:
2054:
2051:
2050:
2005:
2002:
2001:
1985:
1982:
1981:
1955:
1954:
1943:
1940:
1939:
1923:
1909:
1906:
1905:
1902:
1858:
1855:
1854:
1821:
1818:
1817:
1801:
1798:
1797:
1772:
1769:
1768:
1752:
1749:
1748:
1746:random function
1716:
1713:
1712:
1696:
1693:
1692:
1669:
1666:
1665:
1649:
1646:
1645:
1596:
1593:
1592:
1546:
1543:
1542:
1496:
1493:
1492:
1467:
1464:
1463:
1443:
1440:
1439:
1436:data structures
1390:
1387:
1386:
1368:
1365:
1364:
1324:complex numbers
1299:
1296:
1295:
1279:
1276:
1275:
1237:
1229:
1226:
1225:
1218:
1178:
1175:
1174:
1154:
1151:
1150:
1118:
1115:
1114:
1083:
1075:
1072:
1071:
1051:
1048:
1047:
1046:In many cases,
1044:
953:
950:
949:
924:
921:
920:
904:
901:
900:
863:
860:
859:
825:
824:
813:
810:
809:
790:
787:
786:
770:
767:
766:
743:
740:
739:
711:
708:
707:
688:
685:
684:
682:random variable
678:
658:random elements
653:random variable
580:. However, the
541:
538:
537:
521:
518:
517:
486:
483:
482:
455:
452:
451:
435:
432:
431:
403:
400:
399:
360:random quantity
356:random variable
348:
196:Random variable
147:Bernoulli trial
28:
23:
22:
15:
12:
11:
5:
14297:
14287:
14286:
14269:
14268:
14266:
14265:
14253:
14241:
14227:
14214:
14211:
14210:
14207:
14206:
14203:
14202:
14200:
14199:
14194:
14189:
14184:
14179:
14173:
14171:
14165:
14164:
14162:
14161:
14156:
14151:
14146:
14141:
14136:
14131:
14126:
14121:
14116:
14110:
14108:
14102:
14101:
14099:
14098:
14093:
14088:
14079:
14074:
14069:
14063:
14061:
14055:
14054:
14052:
14051:
14046:
14041:
14032:
14030:Bioinformatics
14026:
14024:
14014:
14013:
14001:
14000:
13997:
13996:
13993:
13992:
13989:
13988:
13986:
13985:
13979:
13977:
13973:
13972:
13970:
13969:
13963:
13961:
13955:
13954:
13952:
13951:
13946:
13941:
13936:
13930:
13928:
13919:
13913:
13912:
13909:
13908:
13906:
13905:
13900:
13895:
13890:
13885:
13879:
13877:
13871:
13870:
13868:
13867:
13862:
13857:
13849:
13844:
13839:
13838:
13837:
13835:partial (PACF)
13826:
13824:
13818:
13817:
13815:
13814:
13809:
13804:
13796:
13791:
13785:
13783:
13782:Specific tests
13779:
13778:
13776:
13775:
13770:
13765:
13760:
13755:
13750:
13745:
13740:
13734:
13732:
13725:
13719:
13718:
13716:
13715:
13714:
13713:
13712:
13711:
13696:
13695:
13694:
13684:
13682:Classification
13679:
13674:
13669:
13664:
13659:
13654:
13648:
13646:
13640:
13639:
13637:
13636:
13631:
13629:McNemar's test
13626:
13621:
13616:
13611:
13605:
13603:
13593:
13592:
13568:
13567:
13564:
13563:
13560:
13559:
13557:
13556:
13551:
13546:
13541:
13535:
13533:
13527:
13526:
13524:
13523:
13507:
13501:
13499:
13493:
13492:
13490:
13489:
13484:
13479:
13474:
13469:
13467:Semiparametric
13464:
13459:
13453:
13451:
13447:
13446:
13444:
13443:
13438:
13433:
13428:
13422:
13420:
13414:
13413:
13411:
13410:
13405:
13400:
13395:
13390:
13384:
13382:
13376:
13375:
13373:
13372:
13367:
13362:
13357:
13351:
13349:
13339:
13338:
13335:
13334:
13329:
13323:
13315:
13314:
13311:
13310:
13307:
13306:
13304:
13303:
13302:
13301:
13291:
13286:
13281:
13280:
13279:
13274:
13263:
13261:
13255:
13254:
13251:
13250:
13248:
13247:
13242:
13241:
13240:
13232:
13224:
13208:
13205:(MannâWhitney)
13200:
13199:
13198:
13185:
13184:
13183:
13172:
13170:
13164:
13163:
13161:
13160:
13159:
13158:
13153:
13148:
13138:
13133:
13130:(ShapiroâWilk)
13125:
13120:
13115:
13110:
13105:
13097:
13091:
13089:
13083:
13082:
13080:
13079:
13071:
13062:
13050:
13044:
13042:Specific tests
13038:
13037:
13034:
13033:
13031:
13030:
13025:
13020:
13014:
13012:
13006:
13005:
13003:
13002:
12997:
12996:
12995:
12985:
12984:
12983:
12973:
12967:
12965:
12959:
12958:
12956:
12955:
12954:
12953:
12948:
12938:
12933:
12928:
12923:
12918:
12912:
12910:
12904:
12903:
12901:
12900:
12895:
12894:
12893:
12888:
12887:
12886:
12881:
12866:
12865:
12864:
12859:
12854:
12849:
12838:
12836:
12827:
12821:
12820:
12818:
12817:
12812:
12807:
12806:
12805:
12795:
12790:
12789:
12788:
12778:
12777:
12776:
12771:
12766:
12756:
12751:
12746:
12745:
12744:
12739:
12734:
12718:
12717:
12716:
12711:
12706:
12696:
12695:
12694:
12689:
12679:
12678:
12677:
12667:
12666:
12665:
12655:
12650:
12645:
12639:
12637:
12627:
12626:
12614:
12613:
12610:
12609:
12606:
12605:
12603:
12602:
12597:
12592:
12587:
12581:
12579:
12573:
12572:
12570:
12569:
12564:
12559:
12553:
12551:
12547:
12546:
12544:
12543:
12538:
12533:
12528:
12523:
12518:
12513:
12507:
12505:
12499:
12498:
12496:
12495:
12493:Standard error
12490:
12485:
12480:
12479:
12478:
12473:
12462:
12460:
12454:
12453:
12451:
12450:
12445:
12440:
12435:
12430:
12425:
12423:Optimal design
12420:
12415:
12409:
12407:
12397:
12396:
12384:
12383:
12380:
12379:
12376:
12375:
12373:
12372:
12367:
12362:
12357:
12352:
12347:
12342:
12337:
12332:
12327:
12322:
12317:
12312:
12307:
12302:
12296:
12294:
12288:
12287:
12285:
12284:
12279:
12278:
12277:
12272:
12262:
12257:
12251:
12249:
12243:
12242:
12240:
12239:
12234:
12229:
12223:
12221:
12220:Summary tables
12217:
12216:
12214:
12213:
12207:
12205:
12199:
12198:
12195:
12194:
12192:
12191:
12190:
12189:
12184:
12179:
12169:
12163:
12161:
12155:
12154:
12152:
12151:
12146:
12141:
12136:
12131:
12126:
12121:
12115:
12113:
12107:
12106:
12104:
12103:
12098:
12093:
12092:
12091:
12086:
12081:
12076:
12071:
12066:
12061:
12056:
12054:Contraharmonic
12051:
12046:
12035:
12033:
12024:
12014:
12013:
12001:
12000:
11998:
11997:
11992:
11986:
11983:
11982:
11975:
11974:
11967:
11960:
11952:
11946:
11945:
11932:
11909:
11889:
11888:External links
11886:
11884:
11883:
11877:
11855:
11849:
11830:
11816:
11794:
11788:
11773:
11767:
11751:
11749:
11746:
11743:
11742:
11730:
11704:
11689:
11664:
11657:
11639:
11632:
11612:
11597:
11567:
11543:
11540:on 2005-02-09.
11532:
11508:
11484:
11477:. New Series.
11465:
11450:
11430:
11423:
11401:
11400:
11398:
11395:
11393:
11390:
11388:
11387:
11382:
11377:
11372:
11367:
11365:Random variate
11362:
11357:
11355:Random measure
11352:
11347:
11345:Random element
11342:
11337:
11332:
11327:
11322:
11317:
11312:
11306:
11305:
11304:
11288:
11285:
11268:
11246:
11242:
11211:Main article:
11208:
11205:
11193:
11192:
11181:
11178:
11167:
11164:
11161:
11158:
11155:
11152:
11149:
11146:
11143:
11116:
11113:
11109:measure theory
11101:
11100:
11089:
11085:
11081:
11078:
11075:
11072:
11069:
11066:
11063:
11060:
11057:
11053:
11047:
11043:
11039:
11036:
11033:
11030:
11027:
11024:
11021:
11018:
11013:
11009:
10994:
10993:
10982:
10979:
10976:
10973:
10970:
10967:
10964:
10961:
10958:
10931:
10918:
10910:
10883:
10880:
10864:
10863:
10852:
10849:
10840:
10837:
10834:
10831:
10828:
10825:
10822:
10819:
10816:
10813:
10810:
10807:
10804:
10801:
10798:
10775:
10768:
10763:
10756:
10732:
10729:
10721:
10718:
10717:
10716:
10697:
10686:
10683:
10671:
10670:
10659:
10656:
10651:
10646:
10642:
10638:
10635:
10631:
10625:
10621:
10617:
10614:
10609:
10604:
10601:
10598:
10594:
10590:
10585:
10580:
10576:
10572:
10569:
10565:
10559:
10555:
10551:
10548:
10543:
10538:
10535:
10531:
10527:
10519:
10514:
10510:
10500:
10496:
10492:
10489:
10485:
10480:
10477:
10474:
10471:
10466:
10462:
10447:
10446:
10435:
10427:
10422:
10418:
10413:
10410:
10404:
10401:
10396:
10393:
10390:
10385:
10382:
10377:
10374:
10371:
10367:
10363:
10346:
10345:
10332:
10327:
10324:
10321:
10318:
10315:
10310:
10307:
10302:
10299:
10296:
10292:
10288:
10285:
10271:
10270:
10259:
10255:
10249:
10246:
10241:
10238:
10235:
10230:
10227:
10222:
10218:
10214:
10208:
10204:
10201:
10198:
10195:
10190:
10187:
10182:
10178:
10174:
10169:
10165:
10161:
10157:
10151:
10148:
10143:
10140:
10137:
10132:
10129:
10124:
10120:
10116:
10110:
10106:
10103:
10100:
10097:
10092:
10089:
10084:
10080:
10076:
10071:
10067:
10063:
10060:
10057:
10054:
10049:
10045:
10021:
10001:
9986:
9985:
9974:
9970:
9964:
9961:
9956:
9953:
9950:
9945:
9942:
9937:
9933:
9929:
9923:
9919:
9916:
9913:
9910:
9905:
9902:
9897:
9893:
9889:
9884:
9880:
9874:
9870:
9866:
9863:
9860:
9857:
9852:
9848:
9824:
9819:
9815:
9811:
9808:
9797:
9796:
9785:
9780:
9775:
9771:
9767:
9764:
9760:
9754:
9750:
9746:
9743:
9740:
9737:
9734:
9730:
9721:
9717:
9713:
9710:
9706:
9701:
9698:
9695:
9692:
9687:
9683:
9655:
9643:
9640:
9628:
9627:
9616:
9611:
9607:
9603:
9600:
9596:
9589:
9586:
9583:
9579:
9574:
9566:
9561:
9557:
9550:
9546:
9542:
9539:
9535:
9528:
9525:
9521:
9516:
9513:
9510:
9507:
9504:
9499:
9495:
9480:
9479:
9468:
9460:
9455:
9451:
9446:
9440:
9437:
9432:
9429:
9426:
9421:
9418:
9414:
9410:
9393:
9392:
9379:
9374:
9371:
9368:
9365:
9360:
9357:
9353:
9349:
9346:
9332:
9331:
9320:
9316:
9310:
9307:
9302:
9299:
9296:
9291:
9288:
9284:
9280:
9274:
9270:
9267:
9264:
9261:
9256:
9253:
9249:
9245:
9240:
9236:
9232:
9229:
9226:
9223:
9220:
9215:
9211:
9187:
9167:
9152:
9151:
9140:
9136:
9130:
9127:
9122:
9119:
9116:
9111:
9108:
9103:
9099:
9095:
9089:
9085:
9082:
9079:
9076:
9071:
9068:
9063:
9059:
9055:
9050:
9046:
9040:
9036:
9032:
9029:
9026:
9023:
9018:
9014:
8990:
8985:
8981:
8977:
8974:
8963:
8962:
8951:
8946:
8942:
8936:
8932:
8928:
8924:
8917:
8914:
8910:
8905:
8902:
8899:
8896:
8891:
8887:
8859:
8847:
8844:
8832:
8831:
8816:
8811:
8808:
8805:
8801:
8797:
8794:
8791:
8788:
8786:
8784:
8776:
8772:
8768:
8765:
8760:
8756:
8752:
8749:
8746:
8742:
8737:
8734:
8731:
8728:
8726:
8724:
8716:
8712:
8706:
8703:
8700:
8695:
8691:
8687:
8684:
8681:
8677:
8673:
8670:
8667:
8663:
8658:
8655:
8652:
8649:
8647:
8645:
8642:
8639:
8636:
8633:
8628:
8624:
8620:
8617:
8614:
8611:
8608:
8603:
8599:
8595:
8592:
8589:
8586:
8584:
8582:
8579:
8576:
8571:
8567:
8563:
8562:
8539:
8536:
8525:
8524:
8512:
8509:
8506:
8503:
8500:
8495:
8491:
8487:
8483:
8480:
8477:
8473:
8470:
8467:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8441:
8438:
8434:
8430:
8427:
8424:
8420:
8417:
8414:
8410:
8407:
8404:
8401:
8398:
8395:
8392:
8389:
8386:
8383:
8380:
8377:
8374:
8369:
8365:
8341:
8338:
8333:
8330:
8326:
8322:
8319:
8316:
8312:
8309:
8306:
8302:
8299:
8279:
8276:
8273:
8262:
8261:
8245:
8241:
8235:
8232:
8228:
8224:
8221:
8218:
8214:
8209:
8206:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8182:
8179:
8174:
8170:
8146:
8134:
8131:
8130:
8129:
8118:
8115:
8112:
8100:
8095:
8090:
8087:
8082:
8078:
8074:
8071:
8066:
8061:
8056:
8052:
8048:
8045:
8042:
8039:
8034:
8030:
8015:
8014:
8003:
8000:
7995:
7990:
7987:
7984:
7979:
7974:
7971:
7968:
7965:
7962:
7959:
7954:
7949:
7945:
7941:
7937:
7933:
7930:
7927:
7924:
7921:
7918:
7915:
7910:
7906:
7902:
7899:
7896:
7873:
7870:
7867:
7856:
7855:
7844:
7841:
7838:
7826:
7823:
7820:
7817:
7814:
7809:
7805:
7781:
7778:
7775:
7772:
7769:
7764:
7760:
7756:
7753:
7733:
7730:
7727:
7716:
7715:
7704:
7701:
7698:
7695:
7690:
7686:
7682:
7679:
7676:
7673:
7670:
7667:
7664:
7659:
7655:
7629:
7625:
7621:
7618:
7594:
7582:
7579:
7566:
7546:
7541:
7537:
7533:
7530:
7526:
7522:
7502:
7499:
7496:
7493:
7490:
7466:
7442:
7422:
7419:
7416:
7413:
7410:
7407:
7386:
7382:
7378:
7374:
7371:
7348:
7328:
7301:
7277:
7274:
7271:
7266:
7263:
7258:
7254:
7250:
7245:
7241:
7229:
7228:
7216:
7210:
7207:
7202:
7199:
7196:
7191:
7188:
7183:
7179:
7175:
7169:
7165:
7162:
7159:
7156:
7151:
7148:
7143:
7139:
7135:
7130:
7126:
7120:
7116:
7112:
7109:
7106:
7103:
7098:
7094:
7066:
7061:
7057:
7053:
7050:
7047:
7044:
7022:
7018:
6997:
6977:
6966:
6965:
6954:
6950:
6944:
6941:
6936:
6933:
6930:
6927:
6924:
6918:
6912:
6907:
6904:
6901:
6898:
6893:
6886:
6882:
6878:
6875:
6872:
6869:
6864:
6860:
6836:
6808:
6797:
6796:
6783:
6778:
6768:
6765:
6761:
6757:
6754:
6746:
6744:
6741:
6738:
6735:
6732:
6729:
6726:
6721:
6717:
6713:
6710:
6707:
6704:
6701:
6698:
6695:
6692:
6689:
6686:
6683:
6680:
6677:
6674:
6673:
6670:
6667:
6657:
6654:
6650:
6646:
6643:
6635:
6633:
6630:
6627:
6624:
6621:
6618:
6615:
6610:
6606:
6602:
6599:
6596:
6593:
6590:
6587:
6584:
6581:
6578:
6575:
6572:
6569:
6568:
6566:
6561:
6558:
6555:
6552:
6549:
6546:
6543:
6540:
6537:
6534:
6531:
6528:
6525:
6522:
6519:
6514:
6510:
6478:
6458:
6449:exists, where
6436:
6433:
6429:
6425:
6422:
6402:
6391:
6390:
6379:
6376:
6373:
6370:
6367:
6364:
6361:
6358:
6355:
6352:
6349:
6346:
6343:
6340:
6337:
6332:
6328:
6304:
6280:
6277:
6274:
6271:
6268:
6265:
6245:
6220:
6216:
6212:
6208:
6205:
6182:
6179:
6175:expected value
6162:
6138:
6130:
6127:
6124:
6092:
6089:
6086:
6083:
6080:
6077:
6053:
6029:
6026:
6023:
6020:
6015:
6011:
6007:
6004:
6001:
5981:
5976:
5972:
5968:
5948:
5921:
5901:
5898:
5895:
5892:
5889:
5861:
5841:
5838:
5835:
5832:
5829:
5826:
5823:
5820:
5800:
5797:
5794:
5791:
5788:
5785:
5782:
5779:
5751:
5748:
5745:
5742:
5739:
5728:expected value
5722:
5719:
5706:
5703:
5700:
5697:
5694:
5691:
5688:
5685:
5680:
5677:
5673:
5669:
5666:
5663:
5660:
5657:
5654:
5651:
5648:
5645:
5642:
5639:
5619:
5615:
5611:
5608:
5605:
5602:
5599:
5596:
5593:
5590:
5587:
5584:
5573:
5572:
5561:
5557:
5553:
5550:
5547:
5541:
5536:
5533:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5482:
5478:
5475:
5472:
5469:
5449:
5446:
5443:
5438:
5433:
5430:
5427:
5415:
5412:
5394:
5373:
5351:
5330:
5325:
5320:
5317:
5314:
5294:
5274:
5271:
5268:
5263:
5236:
5207:
5184:
5162:
5140:
5120:
5098:
5076:
5051:
5046:
5043:
5023:
5020:
5017:
5014:
5011:
5008:
5005:
5002:
4999:
4996:
4993:
4990:
4987:
4984:
4979:
4976:
4972:
4949:
4944:
4941:
4938:
4935:
4930:
4927:
4923:
4900:
4872:
4867:
4864:
4844:
4841:
4838:
4835:
4832:
4810:
4805:
4800:
4797:
4794:
4769:
4764:
4759:
4756:
4753:
4729:
4726:
4723:
4718:
4713:
4710:
4707:
4662:measure theory
4653:
4650:
4593:
4590:
4569:
4545:
4522:
4519:
4516:
4513:
4510:
4485:
4480:
4472:
4470:
4467:
4464:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4442:
4436:
4433:
4430:
4426:
4420:
4419:
4417:
4412:
4409:
4406:
4403:
4398:
4394:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4308:
4305:
4302:
4297:
4294:
4291:
4285:
4281:
4277:
4274:
4271:
4268:
4265:
4262:
4257:
4253:
4248:
4244:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4144:
4140:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4067:
4063:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4020:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3886:
3882:
3878:
3875:
3872:
3836:
3833:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3741:
3736:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3699:
3696:
3693:
3690:
3685:
3681:
3666:
3645:
3641:
3637:
3632:
3628:
3624:
3621:
3618:
3613:
3609:
3605:
3600:
3596:
3592:
3589:
3586:
3570:
3563:
3536:
3533:
3519:
3514:
3511:
3508:
3505:
3497:
3495:
3489:
3486:
3480:
3479:
3476:
3473:
3470:
3467:
3459:
3457:
3451:
3448:
3442:
3441:
3439:
3434:
3431:
3428:
3425:
3420:
3416:
3393:
3389:
3353:
3348:
3340:
3337:
3329:
3327:
3324:
3321:
3320:
3317:
3309:
3306:
3298:
3296:
3293:
3290:
3289:
3287:
3282:
3279:
3276:
3273:
3270:
3250:
3230:
3222:
3214:
3211:
3208:
3196:
3193:
3180:
3175:
3171:
3167:
3147:
3144:
3141:
3121:
3118:
3115:
3112:
3109:
3104:
3100:
3079:
3076:
3073:
3053:
3050:
3047:
3044:
3041:
3036:
3032:
3011:
3008:
3005:
2998:
2994:
2989:
2983:
2979:
2973:
2969:
2965:
2962:
2942:
2939:
2934:
2930:
2924:
2920:
2899:
2896:
2891:
2887:
2866:
2861:
2857:
2853:
2850:
2847:
2842:
2838:
2834:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2732:
2729:
2727:
2724:
2706:
2685:
2653:
2624:
2603:
2581:
2577:
2552:
2549:
2545:
2539:
2535:
2531:
2528:
2523:
2519:
2498:
2478:
2456:
2452:
2442:. The measure
2430:
2407:
2403:
2382:
2362:
2342:
2316:
2312:
2296:
2295:
2284:
2281:
2278:
2275:
2272:
2269:
2266:
2263:
2260:
2257:
2254:
2249:
2245:
2218:
2198:
2178:
2158:
2134:
2111:
2108:
2105:
2100:
2096:
2073:
2070:
2067:
2064:
2061:
2058:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
1989:
1969:
1966:
1963:
1958:
1953:
1950:
1947:
1926:
1922:
1919:
1916:
1913:
1901:
1898:
1897:
1896:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1834:
1831:
1828:
1825:
1805:
1785:
1782:
1779:
1776:
1756:
1742:
1726:
1723:
1720:
1700:
1685:
1673:
1653:
1642:
1630:
1627:
1621:
1615:
1609:
1603:
1600:
1580:
1577:
1571:
1565:
1559:
1553:
1550:
1530:
1527:
1521:
1515:
1509:
1503:
1500:
1471:
1447:
1412:random element
1394:
1372:
1316:Boolean values
1303:
1283:
1254:expected value
1240:
1236:
1233:
1217:
1214:
1182:
1158:
1122:
1113:(or range) of
1100:random element
1086:
1082:
1079:
1055:
1043:
1040:
1039:
1038:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
945:is written as
934:
931:
928:
908:
885:
882:
879:
876:
873:
870:
867:
839:
836:
833:
828:
823:
820:
817:
794:
774:
747:
727:
724:
721:
718:
715:
692:
677:
674:
651:. Sometimes a
590:measure theory
562:
561:
545:
525:
505:
502:
499:
496:
493:
490:
471:
459:
439:
419:
416:
413:
410:
407:
398:(e.g. the set
350:
349:
347:
346:
339:
332:
324:
321:
320:
319:
318:
313:
305:
304:
303:
302:
297:
295:Bayes' theorem
292:
287:
282:
277:
269:
268:
267:
266:
261:
256:
251:
243:
242:
241:
240:
239:
238:
233:
228:
226:Observed value
223:
218:
213:
211:Expected value
208:
203:
193:
188:
187:
186:
181:
176:
171:
166:
161:
151:
150:
149:
139:
138:
137:
132:
127:
122:
117:
107:
102:
94:
93:
92:
91:
86:
81:
80:
79:
69:
68:
67:
54:
53:
45:
44:
38:
37:
26:
9:
6:
4:
3:
2:
14296:
14285:
14282:
14281:
14279:
14264:
14263:
14254:
14252:
14251:
14242:
14240:
14239:
14234:
14228:
14226:
14225:
14216:
14215:
14212:
14198:
14195:
14193:
14192:Geostatistics
14190:
14188:
14185:
14183:
14180:
14178:
14175:
14174:
14172:
14170:
14166:
14160:
14159:Psychometrics
14157:
14155:
14152:
14150:
14147:
14145:
14142:
14140:
14137:
14135:
14132:
14130:
14127:
14125:
14122:
14120:
14117:
14115:
14112:
14111:
14109:
14107:
14103:
14097:
14094:
14092:
14089:
14087:
14083:
14080:
14078:
14075:
14073:
14070:
14068:
14065:
14064:
14062:
14060:
14056:
14050:
14047:
14045:
14042:
14040:
14036:
14033:
14031:
14028:
14027:
14025:
14023:
14022:Biostatistics
14019:
14015:
14011:
14006:
14002:
13984:
13983:Log-rank test
13981:
13980:
13978:
13974:
13968:
13965:
13964:
13962:
13960:
13956:
13950:
13947:
13945:
13942:
13940:
13937:
13935:
13932:
13931:
13929:
13927:
13923:
13920:
13918:
13914:
13904:
13901:
13899:
13896:
13894:
13891:
13889:
13886:
13884:
13881:
13880:
13878:
13876:
13872:
13866:
13863:
13861:
13858:
13856:
13854:(BoxâJenkins)
13850:
13848:
13845:
13843:
13840:
13836:
13833:
13832:
13831:
13828:
13827:
13825:
13823:
13819:
13813:
13810:
13808:
13807:DurbinâWatson
13805:
13803:
13797:
13795:
13792:
13790:
13789:DickeyâFuller
13787:
13786:
13784:
13780:
13774:
13771:
13769:
13766:
13764:
13763:Cointegration
13761:
13759:
13756:
13754:
13751:
13749:
13746:
13744:
13741:
13739:
13738:Decomposition
13736:
13735:
13733:
13729:
13726:
13724:
13720:
13710:
13707:
13706:
13705:
13702:
13701:
13700:
13697:
13693:
13690:
13689:
13688:
13685:
13683:
13680:
13678:
13675:
13673:
13670:
13668:
13665:
13663:
13660:
13658:
13655:
13653:
13650:
13649:
13647:
13645:
13641:
13635:
13632:
13630:
13627:
13625:
13622:
13620:
13617:
13615:
13612:
13610:
13609:Cohen's kappa
13607:
13606:
13604:
13602:
13598:
13594:
13590:
13586:
13582:
13578:
13573:
13569:
13555:
13552:
13550:
13547:
13545:
13542:
13540:
13537:
13536:
13534:
13532:
13528:
13522:
13518:
13514:
13508:
13506:
13503:
13502:
13500:
13498:
13494:
13488:
13485:
13483:
13480:
13478:
13475:
13473:
13470:
13468:
13465:
13463:
13462:Nonparametric
13460:
13458:
13455:
13454:
13452:
13448:
13442:
13439:
13437:
13434:
13432:
13429:
13427:
13424:
13423:
13421:
13419:
13415:
13409:
13406:
13404:
13401:
13399:
13396:
13394:
13391:
13389:
13386:
13385:
13383:
13381:
13377:
13371:
13368:
13366:
13363:
13361:
13358:
13356:
13353:
13352:
13350:
13348:
13344:
13340:
13333:
13330:
13328:
13325:
13324:
13320:
13316:
13300:
13297:
13296:
13295:
13292:
13290:
13287:
13285:
13282:
13278:
13275:
13273:
13270:
13269:
13268:
13265:
13264:
13262:
13260:
13256:
13246:
13243:
13239:
13233:
13231:
13225:
13223:
13217:
13216:
13215:
13212:
13211:Nonparametric
13209:
13207:
13201:
13197:
13194:
13193:
13192:
13186:
13182:
13181:Sample median
13179:
13178:
13177:
13174:
13173:
13171:
13169:
13165:
13157:
13154:
13152:
13149:
13147:
13144:
13143:
13142:
13139:
13137:
13134:
13132:
13126:
13124:
13121:
13119:
13116:
13114:
13111:
13109:
13106:
13104:
13102:
13098:
13096:
13093:
13092:
13090:
13088:
13084:
13078:
13076:
13072:
13070:
13068:
13063:
13061:
13056:
13052:
13051:
13048:
13045:
13043:
13039:
13029:
13026:
13024:
13021:
13019:
13016:
13015:
13013:
13011:
13007:
13001:
12998:
12994:
12991:
12990:
12989:
12986:
12982:
12979:
12978:
12977:
12974:
12972:
12969:
12968:
12966:
12964:
12960:
12952:
12949:
12947:
12944:
12943:
12942:
12939:
12937:
12934:
12932:
12929:
12927:
12924:
12922:
12919:
12917:
12914:
12913:
12911:
12909:
12905:
12899:
12896:
12892:
12889:
12885:
12882:
12880:
12877:
12876:
12875:
12872:
12871:
12870:
12867:
12863:
12860:
12858:
12855:
12853:
12850:
12848:
12845:
12844:
12843:
12840:
12839:
12837:
12835:
12831:
12828:
12826:
12822:
12816:
12813:
12811:
12808:
12804:
12801:
12800:
12799:
12796:
12794:
12791:
12787:
12786:loss function
12784:
12783:
12782:
12779:
12775:
12772:
12770:
12767:
12765:
12762:
12761:
12760:
12757:
12755:
12752:
12750:
12747:
12743:
12740:
12738:
12735:
12733:
12727:
12724:
12723:
12722:
12719:
12715:
12712:
12710:
12707:
12705:
12702:
12701:
12700:
12697:
12693:
12690:
12688:
12685:
12684:
12683:
12680:
12676:
12673:
12672:
12671:
12668:
12664:
12661:
12660:
12659:
12656:
12654:
12651:
12649:
12646:
12644:
12641:
12640:
12638:
12636:
12632:
12628:
12624:
12619:
12615:
12601:
12598:
12596:
12593:
12591:
12588:
12586:
12583:
12582:
12580:
12578:
12574:
12568:
12565:
12563:
12560:
12558:
12555:
12554:
12552:
12548:
12542:
12539:
12537:
12534:
12532:
12529:
12527:
12524:
12522:
12519:
12517:
12514:
12512:
12509:
12508:
12506:
12504:
12500:
12494:
12491:
12489:
12488:Questionnaire
12486:
12484:
12481:
12477:
12474:
12472:
12469:
12468:
12467:
12464:
12463:
12461:
12459:
12455:
12449:
12446:
12444:
12441:
12439:
12436:
12434:
12431:
12429:
12426:
12424:
12421:
12419:
12416:
12414:
12411:
12410:
12408:
12406:
12402:
12398:
12394:
12389:
12385:
12371:
12368:
12366:
12363:
12361:
12358:
12356:
12353:
12351:
12348:
12346:
12343:
12341:
12338:
12336:
12333:
12331:
12328:
12326:
12323:
12321:
12318:
12316:
12315:Control chart
12313:
12311:
12308:
12306:
12303:
12301:
12298:
12297:
12295:
12293:
12289:
12283:
12280:
12276:
12273:
12271:
12268:
12267:
12266:
12263:
12261:
12258:
12256:
12253:
12252:
12250:
12248:
12244:
12238:
12235:
12233:
12230:
12228:
12225:
12224:
12222:
12218:
12212:
12209:
12208:
12206:
12204:
12200:
12188:
12185:
12183:
12180:
12178:
12175:
12174:
12173:
12170:
12168:
12165:
12164:
12162:
12160:
12156:
12150:
12147:
12145:
12142:
12140:
12137:
12135:
12132:
12130:
12127:
12125:
12122:
12120:
12117:
12116:
12114:
12112:
12108:
12102:
12099:
12097:
12094:
12090:
12087:
12085:
12082:
12080:
12077:
12075:
12072:
12070:
12067:
12065:
12062:
12060:
12057:
12055:
12052:
12050:
12047:
12045:
12042:
12041:
12040:
12037:
12036:
12034:
12032:
12028:
12025:
12023:
12019:
12015:
12011:
12006:
12002:
11996:
11993:
11991:
11988:
11987:
11984:
11980:
11973:
11968:
11966:
11961:
11959:
11954:
11953:
11950:
11940:
11939:
11933:
11929:
11924:
11917:
11916:
11910:
11906:
11902:
11901:
11896:
11892:
11891:
11880:
11878:0-07-119981-0
11874:
11870:
11866:
11865:
11860:
11856:
11852:
11850:0-387-95313-2
11846:
11842:
11838:
11837:
11831:
11827:
11823:
11819:
11817:0-12-394960-2
11813:
11809:
11805:
11804:
11799:
11795:
11791:
11785:
11781:
11780:
11774:
11770:
11768:3-7643-3807-5
11764:
11760:
11759:
11753:
11752:
11739:
11734:
11715:
11708:
11700:
11696:
11692:
11686:
11682:
11675:
11673:
11671:
11669:
11660:
11658:9781466575592
11654:
11650:
11643:
11635:
11633:9781118344941
11629:
11625:
11624:
11616:
11608:
11604:
11600:
11594:
11590:
11586:
11582:
11578:
11571:
11557:
11553:
11547:
11539:
11535:
11529:
11525:
11521:
11520:
11512:
11498:
11494:
11488:
11480:
11476:
11469:
11461:
11457:
11453:
11447:
11443:
11442:
11434:
11426:
11424:9781466575592
11420:
11417:. CRC Press.
11416:
11409:
11407:
11402:
11386:
11383:
11381:
11378:
11376:
11373:
11371:
11370:Random vector
11368:
11366:
11363:
11361:
11358:
11356:
11353:
11351:
11348:
11346:
11343:
11341:
11338:
11336:
11333:
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11328:
11326:
11323:
11321:
11318:
11316:
11313:
11311:
11308:
11307:
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11296:
11291:
11284:
11282:
11266:
11244:
11240:
11230:
11228:
11224:
11220:
11214:
11204:
11202:
11198:
11197:measure space
11179:
11176:
11172:for all
11162:
11156:
11153:
11147:
11141:
11134:
11133:
11132:
11130:
11126:
11122:
11112:
11110:
11106:
11087:
11076:
11070:
11067:
11061:
11055:
11045:
11037:
11034:
11031:
11025:
11022:
11019:
11007:
10999:
10998:
10997:
10980:
10977:
10971:
10968:
10965:
10959:
10949:
10948:
10947:
10945:
10929:
10916:
10908:
10900:
10899:
10898:almost surely
10893:
10889:
10879:
10877:
10873:
10869:
10850:
10847:
10844:for all
10835:
10832:
10829:
10823:
10817:
10811:
10808:
10805:
10799:
10789:
10788:
10787:
10773:
10766:
10761:
10754:
10746:
10742:
10738:
10728:
10725:
10714:
10713:convex subset
10710:
10706:
10702:
10698:
10695:
10694:
10689:
10688:
10682:
10680:
10676:
10657:
10644:
10640:
10636:
10629:
10623:
10615:
10612:
10607:
10602:
10596:
10592:
10588:
10578:
10574:
10570:
10563:
10557:
10549:
10546:
10541:
10533:
10529:
10517:
10512:
10508:
10498:
10494:
10490:
10487:
10483:
10478:
10472:
10464:
10460:
10452:
10451:
10450:
10433:
10425:
10420:
10416:
10411:
10408:
10402:
10399:
10391:
10383:
10380:
10375:
10372:
10369:
10365:
10361:
10351:
10350:
10349:
10330:
10325:
10322:
10316:
10308:
10305:
10300:
10297:
10294:
10290:
10286:
10283:
10276:
10275:
10274:
10257:
10253:
10247:
10244:
10236:
10228:
10225:
10220:
10216:
10212:
10206:
10196:
10188:
10185:
10180:
10176:
10167:
10163:
10159:
10155:
10149:
10146:
10138:
10130:
10127:
10122:
10118:
10114:
10108:
10098:
10090:
10087:
10082:
10078:
10069:
10065:
10061:
10055:
10047:
10043:
10035:
10034:
10033:
10019:
9999:
9991:
9972:
9968:
9962:
9959:
9951:
9943:
9940:
9935:
9931:
9927:
9921:
9911:
9903:
9900:
9895:
9891:
9882:
9878:
9872:
9868:
9864:
9858:
9850:
9846:
9838:
9837:
9836:
9822:
9817:
9813:
9809:
9806:
9783:
9773:
9769:
9765:
9758:
9752:
9744:
9741:
9738:
9732:
9728:
9719:
9715:
9711:
9708:
9704:
9699:
9693:
9685:
9681:
9673:
9672:
9671:
9669:
9653:
9639:
9637:
9633:
9614:
9609:
9605:
9601:
9598:
9594:
9587:
9584:
9581:
9577:
9572:
9564:
9559:
9555:
9548:
9544:
9540:
9537:
9533:
9526:
9523:
9519:
9514:
9511:
9505:
9497:
9493:
9485:
9484:
9483:
9466:
9458:
9453:
9449:
9444:
9438:
9435:
9427:
9419:
9416:
9412:
9408:
9398:
9397:
9396:
9377:
9372:
9366:
9358:
9355:
9351:
9347:
9344:
9337:
9336:
9335:
9318:
9314:
9308:
9305:
9297:
9289:
9286:
9282:
9278:
9272:
9262:
9254:
9251:
9247:
9238:
9234:
9230:
9227:
9221:
9213:
9209:
9201:
9200:
9199:
9185:
9165:
9157:
9138:
9134:
9128:
9125:
9117:
9109:
9106:
9101:
9097:
9093:
9087:
9077:
9069:
9066:
9061:
9057:
9048:
9044:
9038:
9034:
9030:
9024:
9016:
9012:
9004:
9003:
9002:
8988:
8983:
8979:
8975:
8972:
8949:
8944:
8940:
8934:
8930:
8926:
8922:
8915:
8912:
8908:
8903:
8897:
8889:
8885:
8877:
8876:
8875:
8873:
8857:
8843:
8841:
8837:
8834:which is the
8814:
8809:
8806:
8803:
8799:
8795:
8792:
8789:
8787:
8774:
8766:
8763:
8758:
8754:
8750:
8747:
8740:
8735:
8732:
8729:
8727:
8714:
8701:
8698:
8693:
8689:
8682:
8679:
8675:
8671:
8668:
8661:
8656:
8653:
8650:
8648:
8634:
8631:
8626:
8622:
8615:
8612:
8609:
8601:
8597:
8593:
8590:
8587:
8585:
8577:
8569:
8565:
8553:
8552:
8551:
8537:
8534:
8510:
8501:
8498:
8493:
8489:
8471:
8468:
8465:
8459:
8456:
8450:
8447:
8439:
8436:
8432:
8428:
8425:
8405:
8402:
8396:
8393:
8390:
8384:
8381:
8375:
8367:
8363:
8355:
8354:
8353:
8339:
8331:
8328:
8324:
8320:
8317:
8300:
8297:
8277:
8274:
8271:
8243:
8233:
8230:
8226:
8222:
8219:
8212:
8207:
8201:
8198:
8195:
8189:
8186:
8180:
8172:
8168:
8160:
8159:
8158:
8144:
8116:
8113:
8110:
8093:
8088:
8080:
8076:
8072:
8064:
8054:
8050:
8046:
8040:
8032:
8028:
8020:
8019:
8018:
8001:
7993:
7988:
7985:
7982:
7977:
7972:
7966:
7960:
7952:
7947:
7939:
7928:
7922:
7916:
7913:
7908:
7904:
7897:
7887:
7886:
7885:
7871:
7868:
7865:
7842:
7839:
7836:
7824:
7821:
7815:
7807:
7803:
7795:
7794:
7793:
7779:
7776:
7770:
7767:
7762:
7758:
7751:
7731:
7728:
7725:
7702:
7696:
7693:
7688:
7684:
7677:
7671:
7665:
7657:
7653:
7645:
7644:
7643:
7627:
7623:
7619:
7616:
7608:
7592:
7578:
7564:
7539:
7535:
7531:
7528:
7497:
7494:
7480:
7464:
7456:
7417:
7411:
7408:
7405:
7372:
7369:
7362:
7326:
7318:
7313:
7299:
7291:
7272:
7264:
7261:
7256:
7252:
7248:
7243:
7239:
7214:
7208:
7205:
7197:
7189:
7186:
7181:
7177:
7173:
7167:
7157:
7149:
7146:
7141:
7137:
7128:
7124:
7118:
7114:
7110:
7104:
7096:
7092:
7084:
7083:
7082:
7080:
7059:
7055:
7048:
7045:
7042:
7020:
7016:
6995:
6975:
6952:
6948:
6942:
6939:
6931:
6925:
6922:
6916:
6902:
6896:
6884:
6880:
6876:
6870:
6862:
6858:
6850:
6849:
6848:
6834:
6826:
6822:
6806:
6776:
6766:
6763:
6759:
6755:
6752:
6742:
6733:
6727:
6719:
6715:
6711:
6708:
6705:
6696:
6690:
6687:
6684:
6678:
6665:
6655:
6652:
6648:
6644:
6641:
6631:
6622:
6616:
6608:
6604:
6600:
6591:
6585:
6582:
6579:
6573:
6564:
6559:
6553:
6550:
6544:
6538:
6532:
6526:
6520:
6512:
6508:
6500:
6499:
6498:
6496:
6492:
6476:
6456:
6434:
6431:
6427:
6423:
6420:
6400:
6377:
6371:
6368:
6362:
6356:
6350:
6344:
6338:
6330:
6326:
6318:
6317:
6316:
6302:
6294:
6275:
6269:
6266:
6263:
6243:
6235:
6206:
6203:
6196:
6192:
6188:
6178:
6176:
6160:
6152:
6128:
6125:
6114:
6110:
6106:
6090:
6087:
6081:
6075:
6065:
6051:
6043:
6021:
6013:
6009:
6002:
5974:
5970:
5946:
5938:
5933:
5919:
5896:
5890:
5879:
5875:
5859:
5833:
5827:
5818:
5792:
5786:
5780:
5769:
5767:
5746:
5740:
5729:
5718:
5698:
5695:
5689:
5678:
5675:
5671:
5667:
5661:
5658:
5652:
5646:
5643:
5640:
5609:
5606:
5603:
5597:
5594:
5588:
5559:
5551:
5548:
5534:
5528:
5525:
5519:
5513:
5510:
5507:
5497:
5496:
5495:
5470:
5467:
5444:
5441:
5431:
5411:
5409:
5371:
5363:
5349:
5318:
5315:
5292:
5269:
5252:
5225:
5221:
5205:
5196:
5182:
5138:
5118:
5065:
5044:
5041:
5018:
5015:
5009:
5003:
5000:
4997:
4991:
4985:
4977:
4974:
4970:
4942:
4936:
4928:
4925:
4921:
4913:-measurable;
4888:
4865:
4862:
4842:
4833:
4830:
4822:
4798:
4795:
4783:
4757:
4754:
4743:
4724:
4721:
4711:
4696:
4693:
4691:
4690:intersections
4687:
4683:
4679:
4675:
4674:sigma-algebra
4671:
4667:
4663:
4659:
4649:
4647:
4642:
4630:
4629:
4624:
4623:
4617:
4611:
4607:
4603:
4599:
4589:
4587:
4583:
4559:
4536:
4517:
4514:
4511:
4501:
4500:unit interval
4478:
4468:
4465:
4458:
4455:
4452:
4449:
4446:
4440:
4434:
4431:
4428:
4424:
4415:
4410:
4404:
4396:
4392:
4383:
4379:
4360:
4357:
4354:
4348:
4342:
4339:
4331:
4327:
4322:
4306:
4303:
4300:
4295:
4292:
4289:
4283:
4279:
4272:
4269:
4266:
4260:
4255:
4251:
4246:
4234:
4231:
4227:
4223:
4219:
4214:
4210:
4191:
4188:
4185:
4179:
4173:
4170:
4167:
4142:
4138:
4129:
4125:
4122:is called a "
4106:
4103:
4100:
4094:
4088:
4082:
4076:
4070:
4065:
4061:
4037:
4034:
4031:
4028:
4025:
4022:
4014:
4011:
4005:
3999:
3996:
3993:
3987:
3984:
3977:
3972:
3969:
3968:
3963:
3947:
3945:
3941:
3937:
3933:
3929:
3926:which can be
3925:
3909:
3906:
3900:
3897:
3894:
3884:
3876:
3873:
3862:
3858:
3854:
3850:
3846:
3842:
3832:
3816:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3756:
3750:
3747:
3739:
3734:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3697:
3691:
3683:
3679:
3669:
3665:
3661:
3643:
3639:
3635:
3630:
3626:
3622:
3611:
3607:
3603:
3598:
3594:
3584:
3576:
3569:
3562:
3554:
3550:
3546:
3541:
3532:
3512:
3509:
3506:
3503:
3493:
3487:
3484:
3474:
3471:
3468:
3465:
3455:
3449:
3446:
3437:
3432:
3426:
3418:
3414:
3391:
3387:
3379:
3375:
3371:
3366:
3346:
3338:
3335:
3325:
3322:
3315:
3307:
3304:
3294:
3291:
3285:
3280:
3274:
3268:
3248:
3220:
3209:
3192:
3173:
3169:
3145:
3142:
3139:
3119:
3116:
3110:
3102:
3098:
3077:
3074:
3071:
3051:
3048:
3042:
3034:
3030:
3006:
2996:
2992:
2987:
2981:
2977:
2971:
2967:
2963:
2960:
2940:
2937:
2932:
2928:
2922:
2918:
2897:
2894:
2889:
2885:
2859:
2855:
2848:
2840:
2836:
2823:
2820:
2815:
2801:
2798:
2792:
2786:
2783:
2780:
2774:
2768:
2765:
2762:
2756:
2750:
2747:
2737:
2723:
2721:
2675:
2671:
2667:
2643:
2639:
2601:
2579:
2575:
2566:
2550:
2547:
2543:
2537:
2533:
2529:
2526:
2521:
2517:
2496:
2476:
2454:
2450:
2405:
2401:
2393:to a measure
2360:
2340:
2332:
2314:
2310:
2301:
2279:
2276:
2273:
2267:
2261:
2255:
2247:
2243:
2235:
2234:
2233:
2232:
2216:
2196:
2176:
2156:
2148:
2132:
2123:
2106:
2098:
2094:
2068:
2065:
2062:
2056:
2031:
2028:
2022:
2016:
2013:
2010:
1987:
1961:
1951:
1914:
1911:
1894:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1852:
1851:random vector
1848:
1829:
1823:
1803:
1780:
1774:
1754:
1747:
1743:
1740:
1724:
1721:
1718:
1698:
1690:
1686:
1684:random words.
1671:
1651:
1643:
1625:
1619:
1613:
1607:
1601:
1575:
1569:
1563:
1557:
1551:
1525:
1519:
1513:
1507:
1501:
1489:
1488:
1487:
1485:
1461:
1445:
1437:
1433:
1429:
1425:
1421:
1417:
1413:
1408:
1406:
1392:
1383:
1370:
1363:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1301:
1281:
1274:
1269:
1267:
1263:
1259:
1255:
1234:
1231:
1223:
1213:
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1172:
1156:
1148:
1144:
1140:
1136:
1120:
1112:
1107:
1105:
1101:
1080:
1077:
1069:
1053:
1042:Standard case
1018:
1015:
1009:
1003:
1000:
994:
991:
982:
976:
970:
967:
964:
958:
948:
947:
946:
932:
929:
926:
906:
897:
883:
880:
877:
874:
871:
868:
865:
857:
856:Roman letters
853:
831:
821:
808:
772:
765:
761:
725:
716:
713:
706:
690:
683:
673:
671:
667:
666:George Mackey
664:According to
662:
660:
659:
654:
650:
646:
642:
638:
634:
630:
625:
623:
619:
615:
611:
607:
603:
599:
595:
591:
586:
583:
579:
575:
566:
559:
543:
523:
516:if say heads
500:
497:
494:
491:
480:
476:
472:
457:
437:
414:
411:
408:
397:
393:
389:
385:
384:
383:
381:
377:
373:
369:
365:
361:
358:(also called
357:
345:
340:
338:
333:
331:
326:
325:
323:
322:
317:
314:
312:
309:
308:
307:
306:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
272:
271:
270:
265:
262:
260:
257:
255:
252:
250:
247:
246:
245:
244:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
198:
197:
194:
192:
189:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
156:
155:
152:
148:
145:
144:
143:
140:
136:
133:
131:
128:
126:
123:
121:
118:
116:
113:
112:
111:
108:
106:
103:
101:
98:
97:
96:
95:
90:
87:
85:
84:Indeterminism
82:
78:
75:
74:
73:
70:
66:
63:
62:
61:
58:
57:
56:
55:
51:
47:
46:
43:
40:
39:
36:
32:
31:
19:
14260:
14248:
14229:
14222:
14134:Econometrics
14084: /
14067:Chemometrics
14044:Epidemiology
14037: /
14010:Applications
13852:ARIMA model
13799:Q-statistic
13748:Stationarity
13644:Multivariate
13587: /
13583: /
13581:Multivariate
13579: /
13519: /
13515: /
13289:Bayes factor
13188:Signed rank
13100:
13074:
13066:
13054:
12749:Completeness
12585:Cohort study
12483:Opinion poll
12418:Missing data
12405:Study design
12360:Scatter plot
12282:Scatter plot
12275:Spearman's Ď
12237:Grouped data
11937:
11914:
11898:
11863:
11835:
11802:
11778:
11757:
11733:
11721:. Retrieved
11707:
11680:
11648:
11642:
11622:
11615:
11580:
11570:
11559:. Retrieved
11555:
11546:
11538:the original
11518:
11511:
11500:. Retrieved
11496:
11487:
11478:
11474:
11468:
11440:
11433:
11414:
11310:Aleatoricism
11231:
11216:
11194:
11128:
11124:
11120:
11118:
11102:
10995:
10895:
10891:
10887:
10885:
10865:
10744:
10740:
10736:
10734:
10726:
10723:
10701:vector space
10691:
10672:
10448:
10347:
10272:
9987:
9798:
9645:
9629:
9481:
9394:
9333:
9153:
8964:
8849:
8838:(CDF) of an
8833:
8526:
8263:
8136:
8016:
7857:
7717:
7584:
7314:
7230:
6967:
6798:
6393:If function
6392:
6186:
6184:
6108:
6066:
6042:distribution
5934:
5770:In general,
5763:
5724:
5574:
5417:
5407:
5342:
5197:
5066:
4785:
4697:
4694:
4655:
4643:
4627:
4626:
4621:
4620:
4618:
4597:
4595:
4323:
4235:
4229:
4225:
4221:
4217:
4209:proportional
3973:
3966:
3965:
3961:
3948:
3860:
3857:almost never
3838:
3667:
3663:
3574:
3567:
3560:
3558:
3548:
3544:
3373:
3367:
3198:
2824:
2819:sample space
2816:
2738:
2734:
2670:independence
2297:
2124:
1903:
1893:random field
1689:random graph
1416:graph theory
1409:
1385:
1359:
1270:
1219:
1207:
1194:
1193:is called a
1138:
1108:
1045:
898:
681:
679:
663:
656:
652:
637:real numbers
626:
614:distribution
613:
602:sample space
601:
600:(called the
587:
571:
558:real numbers
396:sample space
372:mathematical
367:
363:
359:
355:
353:
316:Tree diagram
311:Venn diagram
275:Independence
221:Markov chain
195:
105:Sample space
14262:WikiProject
14177:Cartography
14139:Jurimetrics
14091:Reliability
13822:Time domain
13801:(LjungâBox)
13723:Time-series
13601:Categorical
13585:Time-series
13577:Categorical
13512:(Bernoulli)
13347:Correlation
13327:Correlation
13123:JarqueâBera
13095:Chi-squared
12857:M-estimator
12810:Asymptotics
12754:Sufficiency
12521:Interaction
12433:Replication
12413:Effect size
12370:Violin plot
12350:Radar chart
12330:Forest plot
12320:Correlogram
12270:Kendall's Ď
11869:McGrawâHill
11207:Convergence
10693:convolution
6256:. That is,
6234:real-valued
6105:categorical
4604:is neither
4128:subinterval
3863:(formally,
2672:based on a
2145:yields the
2122:for short.
1222:real-valued
1068:real-valued
622:independent
231:Random walk
72:Determinism
60:Probability
14129:Demography
13847:ARMA model
13652:Regression
13229:(Friedman)
13190:(Wilcoxon)
13128:Normality
13118:Lilliefors
13065:Student's
12941:Resampling
12815:Robustness
12803:divergence
12793:Efficiency
12731:(monotone)
12726:Likelihood
12643:Population
12476:Stratified
12428:Population
12247:Dependence
12203:Count data
12134:Percentile
12111:Dispersion
12044:Arithmetic
11979:Statistics
11789:8126517719
11748:Literature
11740:, page 11)
11690:188652940X
11561:2020-08-21
11502:2020-08-21
11460:1104219401
11392:References
11375:Randomness
11201:experiment
10673:This is a
9630:This is a
7035:such that
4784:. Then an
4684:number of
4592:Mixed type
4332:of a CURV
4233:, one has
3845:continuous
3671:given by:
3406:given by:
1264:, and the
1216:Extensions
1104:extensions
676:Definition
382:in which
142:Experiment
89:Randomness
35:statistics
13510:Logistic
13277:posterior
13203:Rank sum
12951:Jackknife
12946:Bootstrap
12764:Bootstrap
12699:Parameter
12648:Statistic
12443:Statistic
12355:Run chart
12340:Pie chart
12335:Histogram
12325:Fan chart
12300:Bar chart
12182:L-moments
12069:Geometric
11928:1307.2968
11905:EMS Press
11723:April 26,
11607:1431-875X
11219:sequences
11177:ω
11163:ω
11148:ω
11077:ω
11068:−
11062:ω
11046:ω
11038:
11012:∞
10969:≠
10960:
10901:(denoted
10833:≤
10824:
10809:≤
10800:
10747:(denoted
10677:with one
10641:σ
10616:μ
10613:−
10603:−
10597:−
10575:σ
10550:μ
10547:−
10534:−
10495:σ
10491:π
10412:±
10381:−
10326:±
10306:−
10226:−
10186:−
10128:−
10088:−
9990:monotonic
9941:−
9901:−
9869:∑
9770:σ
9745:μ
9742:−
9733:−
9716:σ
9712:π
9642:Example 4
9634:with one
9599:−
9585:π
9538:−
9527:π
9417:−
9356:−
9287:−
9252:−
9156:monotonic
9107:−
9067:−
9035:∑
8927:−
8916:π
8846:Example 3
8810:θ
8804:−
8796:−
8775:θ
8764:−
8736:−
8715:θ
8699:−
8683:
8657:−
8632:−
8616:
8610:−
8594:−
8499:−
8472:−
8469:≥
8448:≤
8437:−
8394:≤
8329:−
8272:θ
8244:θ
8231:−
8199:≤
8133:Example 2
8114:≥
8089:−
8073:−
7989:≤
7983:≤
7973:−
7967:
7948:≤
7929:
7914:≤
7898:
7869:≥
7768:≤
7694:≤
7678:
7581:Example 1
7492:Ω
7441:Ω
7381:→
7373::
7347:Ω
7262:−
7187:−
7147:−
7115:∑
6988:but each
6764:−
6712:−
6688:≥
6679:
6653:−
6583:≤
6574:
6551:≤
6533:
6432:−
6369:≤
6351:
6215:→
6207::
6003:
5891:
5828:
5781:
5741:
5693:∞
5690:−
5676:−
5659:≤
5653:ω
5641:ω
5610:∈
5592:∞
5589:−
5552:∈
5546:∀
5535:∈
5526:≤
5520:ω
5508:ω
5477:→
5474:Ω
5471::
5429:Ω
5224:Ď-algebra
5075:Ω
5045:∈
5016:∈
5010:ω
4998:ω
4975:−
4943:∈
4926:−
4866:∈
4840:→
4837:Ω
4834::
4709:Ω
4658:axiomatic
4475:otherwise
4456:≤
4450:≤
4432:−
4349:
4343:∼
4304:−
4293:−
4261:∈
4180:⊆
4095:
4077:
4071:∼
4035:≤
4029:≤
4015:∈
3924:intervals
3877:∈
3871:∀
3853:occurring
3751:∈
3725:−
3713:−
3535:Dice roll
3370:fair coin
3336:ω
3305:ω
3275:ω
3207:Ω
3195:Coin toss
3143:≥
3099:δ
3031:δ
2988:δ
2968:∑
2919:∑
2802:⋯
2787:
2769:
2751:
2684:Ω
2652:Ω
2602:μ
2551:μ
2381:Ω
2277:≤
2268:
2023:ω
2011:ω
1949:Ω
1921:→
1918:Ω
1915::
1873:…
1722:×
1626:⋯
1576:⋯
1526:⋯
1470:Ω
1356:functions
1352:manifolds
1336:sequences
1135:countable
1109:When the
1016:∈
1010:ω
1001:∣
998:Ω
995:∈
992:ω
983:
968:∈
959:
930:⊆
850:(see the
819:Ω
793:Ω
746:Ω
723:→
720:Ω
717::
492:−
450:or tails
135:Singleton
14278:Category
14224:Category
13917:Survival
13794:Johansen
13517:Binomial
13472:Isotonic
13059:(normal)
12704:location
12511:Blocking
12466:Sampling
12345:QâQ plot
12310:Box plot
12292:Graphics
12187:Skewness
12177:Kurtosis
12149:Variance
12079:Heronian
12074:Harmonic
11861:(1965).
11800:(1986).
11699:51441829
11287:See also
11225:and the
11115:Equality
9646:Suppose
8850:Suppose
8137:Suppose
7609:and let
6749:if
6638:if
6315:is then
6191:applying
5874:variance
4962:, where
4887:preimage
4606:discrete
3976:interval
3944:singular
3500:if
3462:if
3332:if
3301:if
2726:Examples
1384:, or an
1332:matrices
1258:variance
1171:interval
858:such as
760:outcomes
392:outcomes
380:function
216:Variance
14250:Commons
14197:Kriging
14082:Process
14039:studies
13898:Wavelet
13731:General
12898:Plug-in
12692:L space
12471:Cluster
12172:Moments
11990:Outline
11907:, 2001
11826:0854102
11524:Freeman
11199:of the
7884:, then
7744:, then
7398:, then
6193:a real
6113:nominal
5721:Moments
5249:is the
4688:and/or
4636:⁄
4382:support
4211:to the
3962:density
3955:⁄
2953:, then
1328:vectors
1266:moments
1173:) then
1070:, i.e.
604:) to a
596:from a
370:) is a
130:Outcome
14119:Census
13709:Normal
13657:Manova
13477:Robust
13227:2-way
13219:1-way
13057:-test
12728:
12305:Biplot
12096:Median
12089:Lehmer
12031:Center
11875:
11847:
11824:
11814:
11786:
11765:
11697:
11687:
11655:
11630:
11605:
11595:
11530:
11458:
11448:
11421:
10896:equal
10449:Then,
9482:Then,
8352:Then,
8264:where
7359:and a
7231:where
6291:. The
5766:moment
5764:first
4885:, its
4686:unions
4213:length
3376:has a
2563:, the
1891:, and
1623:
1617:
1611:
1605:
1573:
1567:
1561:
1555:
1523:
1517:
1511:
1505:
1484:covary
1354:, and
1348:shapes
1224:case (
388:domain
376:random
77:System
65:Axioms
13743:Trend
13272:prior
13214:anova
13103:-test
13077:-test
13069:-test
12976:Power
12921:Pivot
12714:shape
12709:scale
12159:Shape
12139:Range
12084:Heinz
12059:Cubic
11995:Index
11942:(PDF)
11923:arXiv
11919:(PDF)
11717:(PDF)
11129:equal
7792:, so
6133:green
5218:is a
5198:When
4740:be a
4580:on a
3343:tails
3312:heads
3225:tails
3217:heads
2329:. In
1340:trees
1111:image
1102:(see
762:to a
703:is a
477:is a
475:range
394:in a
366:, or
110:Event
13976:Test
13176:Sign
13028:Wald
12101:Mode
12039:Mean
11873:ISBN
11845:ISBN
11812:ISBN
11784:ISBN
11763:ISBN
11725:2013
11695:OCLC
11685:ISBN
11653:ISBN
11628:ISBN
11603:ISSN
11593:ISBN
11528:ISBN
11481:(1).
11456:OCLC
11446:ISBN
11419:ISBN
11127:are
11123:and
10944:zero
10922:a.s.
10894:are
10890:and
10743:are
10739:and
8275:>
7840:<
7729:<
7585:Let
5876:and
4744:and
4698:Let
4666:sets
4608:nor
3849:gaps
3660:fair
3566:and
3132:for
3075:<
3064:for
2910:and
2895:>
1430:and
1362:type
1344:sets
1256:and
631:and
473:the
386:the
13156:BIC
13151:AIC
11585:doi
11042:sup
11035:ess
8680:log
8613:log
8550:so
8017:so
7858:If
7718:If
7513:to
7477:is
7339:on
6489:'s
6469:is
6295:of
4889:is
4614:CDF
4560:of
4207:is
3964:of
3843:is
3704:min
2825:If
2784:PMF
2766:PMF
2748:PMF
2668:or
2614:on
2567:of
2420:on
2373:on
2149:of
2086:or
1691:on
1066:is
647:or
574:die
14280::
11921:,
11903:,
11897:,
11871:.
11843:.
11822:MR
11820:.
11810:.
11693:.
11667:^
11601:.
11591:.
11583:.
11579:.
11554:.
11526:.
11495:.
11454:.
11405:^
11283:.
11229:.
11111:.
10981:0.
10946::
10878:.
10681:.
9638:.
8842:.
8117:0.
8105:if
7843:0.
7831:if
7642:.
7577:.
6064:.
5932:.
5717:.
5410:.
4780:a
4596:A
4243:Pr
4228:â¤
4224:â¤
4220:â¤
3889:Pr
3817:12
3811:11
3805:10
3735:36
3722:13
3372:,
3090:,
2814:.
2302:,
1744:A
1687:A
1591:,
1541:,
1422:,
1418:,
1407:.
1350:,
1346:,
1342:,
1338:,
1334:,
1330:,
1326:,
1322:,
1318:,
896:.
680:A
668:,
661:.
624:.
362:,
354:A
13101:G
13075:F
13067:t
13055:Z
12774:V
12769:U
11971:e
11964:t
11957:v
11925::
11881:.
11853:.
11828:.
11792:.
11771:.
11727:.
11701:.
11661:.
11636:.
11609:.
11587::
11564:.
11505:.
11479:3
11462:.
11427:.
11267:X
11245:n
11241:X
11180:.
11166:)
11160:(
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6339:y
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6331:Y
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3692:S
3689:(
3684:X
3680:f
3668:X
3664:f
3644:2
3640:n
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3631:1
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3608:n
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2253:(
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2217:X
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2099:X
2095:p
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2069:2
2066:=
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2060:(
2057:P
2035:}
2032:2
2029:=
2026:)
2020:(
2017:X
2014::
2008:{
1988:X
1968:)
1965:P
1962:,
1957:F
1952:,
1946:(
1925:R
1912:X
1879:n
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1870:,
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1864:,
1861:1
1833:)
1830:x
1827:(
1824:F
1804:x
1784:)
1781:x
1778:(
1775:F
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1620:0
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1602:0
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1579:)
1570:0
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1502:1
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1371:E
1302:E
1282:E
1239:R
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1181:X
1157:X
1121:X
1085:R
1081:=
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1054:X
1037:.
1025:)
1022:}
1019:S
1013:)
1007:(
1004:X
989:{
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980:P
977:=
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965:X
962:(
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933:E
927:S
907:X
884:T
881:,
878:Z
875:,
872:Y
869:,
866:X
838:)
835:P
832:,
827:F
822:,
816:(
773:E
726:E
714:X
691:X
560:.
544:T
524:H
504:}
501:1
498:,
495:1
489:{
458:T
438:H
418:}
415:T
412:,
409:H
406:{
343:e
336:t
329:v
20:)
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