3001:
553:. Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed a small circle of mutual scientific friends in Paris about it.
4219:
3814:
91:
13992:
4214:{\displaystyle \operatorname {E} ={\begin{cases}\operatorname {E} -\operatorname {E} &{\text{if }}\operatorname {E} <\infty {\text{ and }}\operatorname {E} <\infty ;\\+\infty &{\text{if }}\operatorname {E} =\infty {\text{ and }}\operatorname {E} <\infty ;\\-\infty &{\text{if }}\operatorname {E} <\infty {\text{ and }}\operatorname {E} =\infty ;\\{\text{undefined}}&{\text{if }}\operatorname {E} =\infty {\text{ and }}\operatorname {E} =\infty .\end{cases}}}
627:... this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for the sum hoped for. We will call this advantage
12406:
1108:
580:
proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from the same principle. But finally I have found that my answers in many cases do not differ from theirs.
2183:
5737:
3737:
568:" on probability theory just after visiting Paris. The book extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the
9126:
1374:
8387:
of the expected value. However, in special cases the Markov and
Chebyshev inequalities often give much weaker information than is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds
579:
It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by
1955:
792:
As discussed above, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably
6324:
5384:
2714:
which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values, the
Lebesgue theory of expectation is identical to the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density
2545:
4380:. The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references.
1752:
illustrates that the value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely.
5591:
3550:
8383:. These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two
4258:
is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations.
545:
in 1654. MΓ©rΓ© claimed that this problem could not be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all.
8934:
1227:
3501:
2701:
Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the
Lebesgue integral of
8857:
8673:
1592:
5919:
2989:
6330:, this means that the expected value of the sum of any finite number of random variables is the sum of the expected values of the individual random variables, and the expected value scales linearly with a multiplicative constant. Symbolically, for
3254:
8373:
11961:
6190:
7498:
3365:
9612:
12816:
9403:
610:
That any one Chance or
Expectation to win any thing is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal chance of gaining them, my Expectation is worth
11568:
446:
can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality.
3746:
There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the
Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any
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5528:
4793:
2363:
11802:
12546:
9933:
5197:
12691:
10494:
8254:
5259:
536:
between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to
12341:
7868:
6559:
1017:
8522:
inputs under-estimates the same weighted average of the two outputs; Jensen's inequality extends this to the setting of completely general weighted averages, as represented by the expectation. In the special case that
2699:
7793:
11106:
2305:
10637:
8497:
7362:
2178:{\displaystyle \operatorname {E} \,=\sum _{i}x_{i}p_{i}=1({\tfrac {c}{2}})+2({\tfrac {c}{8}})+3({\tfrac {c}{24}})+\cdots \,=\,{\tfrac {c}{2}}+{\tfrac {c}{4}}+{\tfrac {c}{8}}+\cdots \,=\,c\,=\,{\tfrac {1}{\ln 2}}.}
1701:
8114:
7583:
5076:
4948:
1441:
game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable
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10326:
7211:
11332:
11175:
7102:
A well defined expectation implies that there is one number, or rather, one constant that defines the expected value. Thus follows that the expectation of this constant is just the original expected value.
2805:
1760:, which implies that the infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable
7678:
6712:
12043:
450:
The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by
9198:
5732:{\displaystyle \int _{k}^{\infty }\alpha k^{\alpha }x^{-\alpha }\,dx={\begin{cases}{\frac {\alpha k}{\alpha -1}}&{\text{if }}\alpha >1\\\infty &{\text{if }}0<\alpha \leq 1\end{cases}}}
11241:
7100:
3374:
6938:
5810:
2334:
Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of
8723:
8553:
1481:
6195:
2902:
1623:
of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that
11639:
1860:
5585:
11044:
12374:
12157:
serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being
4862:
4708:
3159:
11003:
8276:
10818:
10377:
6073:
5253:
11854:
10718:
3732:{\displaystyle \operatorname {E} =\sum _{i=1}^{\infty }x_{i}\,p_{i}=2\cdot {\frac {1}{2}}+4\cdot {\frac {1}{4}}+8\cdot {\frac {1}{8}}+16\cdot {\frac {1}{16}}+\cdots =1+1+1+1+\cdots .}
12238:
of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the
10550:
10197:
7379:
4500:
3259:
3094:
6006:
2995:. The above discussion of continuous random variables is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable.
12189:
that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the
7015:
6827:
6426:
1950:
560:
book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see
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10034:
9320:
5816:
12120:
11845:
11459:
9800:
7625:
6659:
6627:
6116:
10915:
10145:
9434:
10864:
1907:
12401:
774:
13942:
12616:
9121:{\displaystyle {\Bigl (}\operatorname {E} |X+Y|^{p}{\Bigr )}^{1/p}\leq {\Bigl (}\operatorname {E} |X|^{p}{\Bigr )}^{1/p}+{\Bigl (}\operatorname {E} |Y|^{p}{\Bigr )}^{1/p}.}
4443:
11648:
11454:
5433:
4997:
4549:
1369:{\displaystyle \operatorname {E} =1\cdot {\frac {1}{6}}+2\cdot {\frac {1}{6}}+3\cdot {\frac {1}{6}}+4\cdot {\frac {1}{6}}+5\cdot {\frac {1}{6}}+6\cdot {\frac {1}{6}}=3.5.}
797:, as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of
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512:
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10673:
9975:
9825:
12724:
12584:
10948:
10777:
10256:
9490:
9231:
11365:
6561:
If we think of the set of random variables with finite expected value as forming a vector space, then the linearity of expectation implies that the expected value is a
10744:
10223:
10063:
9315:
10386:
9820:
8191:
6773:
6592:
6035:
5967:
1803:
6747:
12079:
11404:
10090:
9495:
7980:
7928:
6375:
62:
11599:
7279:
3114:
12729:
12242:(the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the
10328:
Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let
6967:
6882:
6853:
912:
9635:
9457:
7897:
6186:
6163:
12886:
11424:
9286:
9251:
8154:
8134:
8024:
8004:
7948:
7718:
7698:
7144:
7035:
6348:
6140:
3154:
3134:
3056:
3033:
1460:
1421:
1397:
1222:
1186:
1158:
1134:
3755:. The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable
2646:
9640:
5439:
2232:
10555:
8431:
7292:
6940:
In other words, if X and Y are random variables that take different values with probability zero, then the expectation of X will equal the expectation of Y.
4714:
1626:
8029:
7511:
6319:{\displaystyle {\begin{aligned}\operatorname {E} &=\operatorname {E} +\operatorname {E} ,\\\operatorname {E} &=a\operatorname {E} ,\end{aligned}}}
5128:
12621:
8157:
5379:{\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\int _{-\infty }^{\infty }x\,e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}\,dx=\mu }
1741:
are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context.
382:
3751:
random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then the expected value can be defined as
3511:
Expected values as defined above are automatically finite numbers. However, in many cases it is fundamental to be able to consider expected values of
3000:
13935:
13139:
12268:
2737:
7802:
1744:
However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the
13143:
6431:
11966:
2540:{\displaystyle \int _{a}^{b}xf(x)\,dx=\int _{a}^{b}{\frac {x}{x^{2}+\pi ^{2}}}\,dx={\frac {1}{2}}\ln {\frac {b^{2}+\pi ^{2}}{a^{2}+\pi ^{2}}}.}
7727:
11051:
3547:
ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has
1111:
An illustration of the convergence of sequence averages of rolls of a dice to the expected value of 3.5 as the number of rolls (trials) grows
13827:. Wiley Series in Probability and Mathematical Statistics (Second edition of 1970 original ed.). New York: John Wiley & Sons, Inc.
14094:
14057:
13928:
13727:. Wiley Series in Probability and Mathematical Statistics (Third edition of 1979 original ed.). New York: John Wiley & Sons, Inc.
12179:
2323:, and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard
9233:
pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let
5003:
12956:
12234:
To empirically estimate the expected value of a random variable, one repeatedly measures observations of the variable and computes the
4868:
1462:
represents the (monetary) outcome of a $ 1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability
13586:
13584:
12178:, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their
4555:
10261:
10095:
A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below.
7149:
11253:
11113:
793:
many possible outcomes. It is also very common to consider the distinct case of random variables dictated by (piecewise-)continuous
13581:
12409:
The mass of probability distribution is balanced at the expected value, here a Beta(Ξ±,Ξ²) distribution with expected value Ξ±/(Ξ±+Ξ²).
7630:
2331:
are defined as those corresponding to this special class of densities, although the term is used differently by various authors.
6664:
665:
When "E" is used to denote "expected value", authors use a variety of stylizations: the expectation operator can be stylized as
375:
9146:
7364:
where the values on both sides are well defined or not well defined simultaneously, and the integral is taken in the sense of
13902:
12555:
11427:
11181:
7040:
8675:
This can also be proved by the HΓΆlder inequality. In measure theory, this is particularly notable for proving the inclusion
6890:
5750:
2597:
left undefined otherwise. However, measure-theoretic notions as given below can be used to give a systematic definition of
1478:
in
American roulette), the payoff is $ 35; otherwise the player loses the bet. The expected profit from such a bet will be
808:
Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a
7873:
7796:
2896:
2895:(relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the
11604:
9135:, and are often given in that context. By contrast, the Jensen inequality is special to the case of probability spaces.
12239:
1808:
1099:. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.
13866:
13832:
13766:
13732:
13696:
13451:
13028:
12996:
5541:
4954:
368:
356:
315:
17:
11008:
13975:
12346:
12185:
It is possible to construct an expected value equal to the probability of an event by taking the expectation of an
7282:
4806:
4673:
3036:
2818:
14062:
12457:
Expected values can also be used to compute the variance, by means of the computational formula for the variance
246:
182:
31:
10956:
3496:{\displaystyle \operatorname {E} =\int _{0}^{\infty }{\bigl (}1-F(x){\bigr )}\,dx-\int _{-\infty }^{0}F(x)\,dx,}
13816:
10782:
10331:
8262:
is any random variable with finite expectation, then Markov's inequality may be applied to the random variable
6040:
5210:
294:
155:
10678:
8852:{\displaystyle \operatorname {E} |XY|\leq (\operatorname {E} |X|^{p})^{1/p}(\operatorname {E} |Y|^{q})^{1/q}.}
8668:{\displaystyle \left(\operatorname {E} |X|^{s}\right)^{1/s}\leq \left(\operatorname {E} |X|^{t}\right)^{1/t}.}
1587:{\displaystyle \operatorname {E} =-\$ 1\cdot {\frac {37}{38}}+\$ 35\cdot {\frac {1}{38}}=-\$ {\frac {1}{19}}.}
14099:
10869:
10504:
10150:
8879:
4449:
4357:
3061:
5972:
5914:{\displaystyle {\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {\gamma x}{(x-x_{0})^{2}+\gamma ^{2}}}\,dx}
2984:{\displaystyle \operatorname {E} \equiv \int _{\Omega }X\,d\operatorname {P} =\int _{\mathbb {R} }xf(x)\,dx}
528:
The idea of the expected value originated in the middle of the 17th century from the study of the so-called
13969:
10380:
10099:
7721:
6979:
6781:
6380:
4348:
with remaining probability. Using the definition for non-negative random variables, one can show that both
2198:
1912:
794:
606:
Neither Pascal nor
Huygens used the term "expectation" in its modern sense. In particular, Huygens writes:
13399:
5089:
9980:
8500:
7365:
3760:
1431:
12084:
11809:
9757:
7595:
6632:
6600:
6089:
14051:
12228:
10875:
10105:
9408:
8169:
7231:
150:
13079:
George Mackey (July 1980). "HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY - A HISTORICAL SURVEY".
8514:
has finite expectation, so that the right-hand side is well-defined (possibly infinite). Convexity of
1865:
1756:
For this reason, many mathematical textbooks only consider the case that the infinite sum given above
13963:
13875:
12379:
10826:
8271:
7246:
3249:{\displaystyle x\leq \mu ,\;\,0\leq y\leq F(x)\quad {\text{or}}\quad x\geq \mu ,\;\,F(x)\leq y\leq 1}
2589:
To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral
752:
266:
12593:
8692:
5969:
is true almost surely, when the probability measure attributes zero-mass to the complementary event
5651:
3841:
13952:
12931:
12848:
8368:{\displaystyle \operatorname {P} (|X-{\text{E}}|\geq a)\leq {\frac {\operatorname {Var} }{a^{2}}},}
5935:
The basic properties below (and their names in bold) replicate or follow immediately from those of
5082:
4404:
4377:
2883:
These conditions are all equivalent, although this is nontrivial to establish. In this definition,
325:
320:
209:
194:
13761:. Duxbury Advanced Series (Second edition of 1990 original ed.). Pacific Grove, CA: Duxbury.
12253:
This property is often exploited in a wide variety of applications, including general problems of
11432:
5397:
4961:
4513:
676:
495:
12837:
12832:
11956:{\displaystyle \operatorname {E} ={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,}
10645:
9940:
644:
304:
175:
13803:(Second edition of 1966 original ed.). New YorkβLondonβSydney: John Wiley & Sons, Inc.
12700:
12560:
10920:
10749:
10228:
9462:
9203:
7493:{\displaystyle \operatorname {E} =\int _{0}^{\infty }(1-F(x))\,dx-\int _{-\infty }^{0}F(x)\,dx,}
3360:{\displaystyle \int _{-\infty }^{\mu }F(x)\,dx=\int _{\mu }^{\infty }{\big (}1-F(x){\big )}\,dx}
13783:(Third edition of 1950 original ed.). New YorkβLondonβSydney: John Wiley & Sons, Inc.
12194:
11337:
8403:
8173:
6327:
4799:
4397:
3516:
1745:
199:
12845:β related to expectations in a way analogous to that in which quantiles are related to medians
10723:
10202:
10039:
9607:{\displaystyle \operatorname {E} =n\cdot \Pr \left(U\in \left\right)=n\cdot {\tfrac {1}{n}}=1}
9291:
7500:
with the integrals taken in the sense of
Lebesgue. As a special case, for any random variable
30:
This article is about the term used in probability theory and statistics. For other uses, see
13099:
12894:
12694:
12227:. The moments of some random variables can be used to specify their distributions, via their
12213:
12150:
12146:
11247:
10950:
9805:
8396:
8389:
6752:
6571:
6014:
5946:
4506:
2826:
1775:
1749:
569:
340:
299:
204:
170:
12963:
12811:{\displaystyle (\Delta A)^{2}=\langle {\hat {A}}^{2}\rangle -\langle {\hat {A}}\rangle ^{2}}
9398:{\displaystyle X_{n}=n\cdot \mathbf {1} \left\{U\in \left(0,{\tfrac {1}{n}}\right)\right\},}
6717:
13912:
13842:
13808:
13788:
13742:
13346:
Characterization of the expected value on the graph of the cumulative distribution function
13339:
12205:
12190:
12158:
12052:
11563:{\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.}
11382:
10068:
8885:
7953:
7901:
7369:
6353:
5943:"βa central property of the Lebesgue integral. Basically, one says that an inequality like
4666:
2641:
2590:
2320:
2312:
1757:
805:, which provide these different contexts with an axiomatic foundation and common language.
802:
616:
459:
430:
330:
224:
117:
65:
40:
13383:
13228:
11575:
7255:
3099:
8:
13850:
13720:
13707:
12865:
12414:
12262:
11642:
7123:
6946:
6861:
6832:
5743:
5534:
5203:
2728:
2339:
2324:
2319:
is described in the next section. The density functions of many common distributions are
1427:
289:
231:
219:
214:
13295:
9617:
9439:
7879:
6168:
6145:
14023:
13854:
13487:
13133:
13061:
12913:β an equation for calculating the expected value of a random number of random variables
12910:
12871:
12186:
12134:
The expectation of a random variable plays an important role in a variety of contexts.
12123:
11409:
9745:{\displaystyle \lim _{n\to \infty }\operatorname {E} =1\neq 0=\operatorname {E} \left.}
9256:
9236:
8384:
8139:
8119:
8009:
7989:
7933:
7703:
7683:
7227:
7129:
7020:
6333:
6125:
5922:
5523:{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }xe^{-x^{2}/2}\,dx=0}
3139:
3119:
3041:
3018:
1445:
1406:
1382:
1207:
1171:
1143:
1119:
557:
396:
276:
165:
105:
82:
13920:
13540:
4788:{\displaystyle \sum _{i=0}^{\infty }{\frac {ie^{-\lambda }\lambda ^{i}}{i!}}=\lambda }
13981:
13898:
13862:
13828:
13762:
13728:
13692:
13121:
13024:
12992:
12551:
12254:
12169:
12046:
11797:{\displaystyle \operatorname {E} ={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\leftdx.}
8686:
5936:
3368:
2626:
591:
529:
335:
241:
140:
12541:{\displaystyle \operatorname {Var} (X)=\operatorname {E} -(\operatorname {E} )^{2}.}
9928:{\displaystyle \operatorname {E} \left\neq \sum _{n=0}^{\infty }\operatorname {E} .}
5192:{\displaystyle \int _{0}^{\infty }\lambda xe^{-\lambda x}\,dx={\frac {1}{\lambda }}}
13890:
13754:
13353:
13053:
13016:
12827:
12258:
10498:
2837:
2206:
1045:
595:
550:
435:
160:
90:
12686:{\displaystyle \langle {\hat {A}}\rangle =\langle \psi |{\hat {A}}|\psi \rangle .}
4367:
Similarly, the Cauchy distribution, as discussed above, has undefined expectation.
13908:
13882:
13838:
13804:
13784:
13738:
12905:
12900:
12265:, since most quantities of interest can be written in terms of expectation, e.g.
12235:
12175:
8421:
7592:
Non-multiplicativity: In general, the expected value is not multiplicative, i.e.
6119:
5390:
2622:
1400:
1096:
696:
542:
443:
439:
236:
187:
13471:
10489:{\displaystyle \operatorname {E} \left=\sum _{i=0}^{\infty }\operatorname {E} .}
8249:{\displaystyle \operatorname {P} (X\geq a)\leq {\frac {\operatorname {E} }{a}}.}
1188:
are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of
594:
became the first person to think systematically in terms of the expectations of
14008:
13796:
13776:
13750:
13435:
13419:
13367:
12418:
11848:
8395:
The following three inequalities are of fundamental importance in the field of
2614:
2308:
1161:
798:
455:
3785:. These are nonnegative random variables, and it can be directly checked that
14088:
13503:
13115:
12587:
12336:{\displaystyle \operatorname {P} ({X\in {\mathcal {A}}})=\operatorname {E} ,}
9132:
8392:
extends the
Chebyshev inequality to the context of sums of random variables.
7983:
6969:
6884:
6594:
5940:
2214:
1620:
1424:
826:
809:
647:
in 1901. The symbol has since become popular for
English writers. In German,
538:
124:
13894:
13044:
Ore, Oystein (1960). "Ore, Pascal and the Invention of Probability Theory".
7863:{\displaystyle \operatorname {E} \neq \operatorname {E} \operatorname {E} ,}
6554:{\textstyle \operatorname {E} \left=\sum _{i=1}^{N}a_{i}\operatorname {E} .}
1012:{\displaystyle \operatorname {E} =x_{1}p_{1}+x_{2}p_{2}+\cdots +x_{k}p_{k}.}
13991:
1952:
is the scaling factor which makes the probabilities sum to 1. Then we have
1614:
1074:
351:
261:
145:
13125:
10552:
be a sequence of non-negative random variables. Fatou's lemma states that
2613:
All definitions of the expected value may be expressed in the language of
13820:
13341:
Charakterisierung des Erwartungswertes am Graphen der Verteilungsfunktion
12550:
A very important application of the expectation value is in the field of
12243:
12154:
11847:
is finite, changing the order of integration, we get, in accordance with
6562:
4376:
The following table gives the expected values of some commonly occurring
2694:{\displaystyle \operatorname {E} =\int _{\Omega }X\,d\operatorname {P} .}
2307:
A general and mathematically precise formulation of this definition uses
271:
112:
100:
13358:
13013:
History of Probability and Statistics and Their Applications before 1750
12440:
Now consider a weightless rod on which are placed weights, at locations
7788:{\displaystyle \operatorname {E} =\operatorname {E} \operatorname {E} .}
14003:
13065:
13020:
12162:
12138:
11101:{\displaystyle \operatorname {E} |X|\leq \operatorname {E} <\infty }
2300:{\displaystyle \operatorname {E} =\int _{-\infty }^{\infty }xf(x)\,dx.}
129:
75:
13170:
10632:{\displaystyle \operatorname {E} \leq \liminf _{n}\operatorname {E} .}
8492:{\displaystyle f(\operatorname {E} (X))\leq \operatorname {E} (f(X)).}
7357:{\displaystyle \operatorname {E} =\int _{-\infty }^{\infty }x\,dF(x),}
7245:. This is nothing but a different way of stating the expectation of a
14073:
13889:(Twelfth edition of 1972 original ed.). London: Academic Press.
13801:
An introduction to probability theory and its applications. Volume II
13569:
12842:
12142:
9138:
6714:
Proof follows from the linearity and the non-negativity property for
2808:
1696:{\displaystyle \operatorname {E} =\sum _{i=1}^{\infty }x_{i}\,p_{i},}
13781:
An introduction to probability theory and its applications. Volume I
13319:
13307:
13285:
13283:
13057:
12405:
8518:
can be phrased as saying that the output of the weighted average of
8172:
control the likelihood of a random variable taking on large values.
8109:{\displaystyle \operatorname {E} =\int _{\mathbb {R} }g(x)f(x)\,dx.}
7578:{\displaystyle \operatorname {E} =\sum _{n=0}^{\infty }\Pr(X>n),}
14068:
14038:
14033:
14028:
14018:
13861:(Fourth edition of 1965 original ed.). New York: McGraw-Hill.
12247:
8680:
8380:
2218:
1438:
887:
of possible outcomes, each of which (respectively) has probability
549:
He began to discuss the problem in the famous series of letters to
451:
256:
5071:{\displaystyle \int _{a}^{b}{\frac {x}{b-a}}\,dx={\frac {a+b}{2}}}
3519:, in which one considers a random variable with possible outcomes
13663:
13280:
13218:
13216:
9131:
The HΓΆlder and Minkowski inequalities can be extended to general
4943:{\displaystyle \sum _{i=1}^{\infty }ip(1-p)^{i-1}={\frac {1}{p}}}
13689:
Pascal's arithmetical triangle: the story of a mathematical idea
13641:
13639:
13637:
13635:
4655:{\displaystyle \sum _{i=0}^{n}i{n \choose i}p^{i}(1-p)^{n-i}=np}
13100:"The Value of Chances in Games of Fortune. English Translation"
10321:{\displaystyle \lim _{n}\operatorname {E} =\operatorname {E} .}
7206:{\displaystyle |\operatorname {E} |\leq \operatorname {E} |X|.}
1107:
856:
623:", where the concept of expected value was defined explicitly:
13651:
13620:
13608:
13515:
13213:
12454:(whose sum is one). The point at which the rod balances is E.
12161:; that is, the expected value of the estimate is equal to the
11327:{\displaystyle \lim _{n}\operatorname {E} =\operatorname {E} }
11170:{\displaystyle \lim _{n}\operatorname {E} =\operatorname {E} }
10258:
pointwise. Then, the monotone convergence theorem states that
7870:
although in special cases of dependency the equality may hold.
1610:. Thus, in 190 bets, the net loss will probably be about $ 10.
13632:
13557:
13244:
13201:
12421:
is an analogous concept to expectation. For example, suppose
2800:{\displaystyle \operatorname {P} (X\in A)=\int _{A}f(x)\,dx,}
12989:
All of Statistics: a concise course in statistical inference
11374:
3371:
converge. Finally, this is equivalent to the representation
7673:{\displaystyle \operatorname {E} \cdot \operatorname {E} .}
5725:
4207:
1594:
That is, the expected value to be won from a $ 1 bet is β$
1377:
1165:
1137:
13268:
12851:β the expected value of the conditional expected value of
6707:{\displaystyle \operatorname {E} \leq \operatorname {E} .}
4371:
454:. In the axiomatic foundation for probability provided by
13189:
12261:, to estimate (probabilistic) quantities of interest via
12038:{\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx}
2608:
13171:"Earliest uses of symbols in probability and statistics"
9193:{\displaystyle \operatorname {E} \to \operatorname {E} }
1619:
Informally, the expectation of a random variable with a
1615:
Random variables with countably infinitely many outcomes
13950:
13859:
Probability, random variables, and stochastic processes
13815:
13509:
13116:
Laplace, Pierre Simon, marquis de, 1749-1827. (1952) .
11236:{\displaystyle \lim _{n}\operatorname {E} |X_{n}-X|=0.}
8116:
This formula also holds in multidimensional case, when
7095:{\displaystyle \operatorname {E} ]=\operatorname {E} .}
3807:
are both then defined as either nonnegative numbers or
11046:
Then, according to the dominated convergence theorem,
10829:
10379:
be non-negative random variables. It follows from the
9754:
Analogously, for general sequence of random variables
9587:
9556:
9371:
6933:{\displaystyle \operatorname {E} =\operatorname {E} .}
6434:
5805:{\displaystyle X\sim \mathrm {Cauchy} (x_{0},\gamma )}
3015:
The expected value of any real-valued random variable
2153:
2122:
2107:
2092:
2066:
2042:
2018:
1923:
1826:
13177:
12874:
12732:
12703:
12624:
12596:
12563:
12463:
12382:
12349:
12271:
12087:
12055:
11969:
11857:
11812:
11651:
11607:
11578:
11462:
11435:
11412:
11385:
11340:
11256:
11184:
11116:
11054:
11011:
10959:
10923:
10878:
10785:
10752:
10726:
10681:
10648:
10558:
10507:
10389:
10334:
10264:
10231:
10205:
10153:
10108:
10071:
10042:
9983:
9943:
9828:
9808:
9760:
9643:
9620:
9498:
9465:
9442:
9411:
9323:
9294:
9259:
9239:
9206:
9149:
8937:
8726:
8556:
8434:
8279:
8194:
8176:
is among the best-known and simplest to prove: for a
8142:
8122:
8032:
8012:
7992:
7956:
7936:
7904:
7882:
7805:
7730:
7706:
7686:
7633:
7598:
7514:
7382:
7295:
7258:
7152:
7132:
7043:
7023:
6982:
6949:
6893:
6864:
6835:
6784:
6755:
6720:
6667:
6635:
6603:
6574:
6383:
6356:
6336:
6193:
6171:
6148:
6128:
6092:
6043:
6017:
5975:
5949:
5819:
5753:
5594:
5544:
5442:
5400:
5262:
5213:
5131:
5092:
5006:
4964:
4871:
4809:
4717:
4676:
4558:
4516:
4452:
4407:
3817:
3553:
3515:. This is intuitive, for example, in the case of the
3377:
3262:
3162:
3142:
3122:
3102:
3064:
3044:
3021:
2905:
2740:
2649:
2366:
2360:. It is straightforward to compute in this case that
2235:
1958:
1915:
1868:
1811:
1778:
1629:
1484:
1448:
1409:
1385:
1230:
1210:
1174:
1146:
1122:
915:
755:
679:
498:
43:
13015:. Wiley Series in Probability and Statistics. 1990.
3739:
It is natural to say that the expected value equals
1136:
represent the outcome of a roll of a fair six-sided
825:. Similarly, one may define the expected value of a
13256:
4263:In the case of the St. Petersburg paradox, one has
1073:In the special case that all possible outcomes are
12880:
12810:
12718:
12685:
12610:
12578:
12556:expectation value of a quantum mechanical operator
12540:
12395:
12368:
12335:
12114:
12073:
12037:
11955:
11839:
11796:
11634:{\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }}
11633:
11593:
11562:
11448:
11418:
11398:
11359:
11326:
11235:
11169:
11100:
11038:
10997:
10942:
10909:
10858:
10812:
10771:
10738:
10712:
10667:
10631:
10544:
10488:
10371:
10320:
10250:
10217:
10191:
10139:
10084:
10057:
10028:
9969:
9927:
9814:
9794:
9744:
9629:
9606:
9484:
9451:
9428:
9397:
9309:
9280:
9245:
9225:
9192:
9139:Expectations under convergence of random variables
9120:
8851:
8667:
8491:
8367:
8248:
8148:
8128:
8108:
8018:
7998:
7974:
7942:
7922:
7891:
7862:
7787:
7712:
7692:
7672:
7619:
7577:
7492:
7356:
7273:
7205:
7138:
7094:
7029:
7009:
6961:
6932:
6876:
6847:
6821:
6767:
6741:
6706:
6653:
6621:
6586:
6553:
6420:
6369:
6342:
6318:
6180:
6157:
6134:
6110:
6067:
6029:
6000:
5961:
5913:
5804:
5731:
5579:
5522:
5427:
5378:
5247:
5191:
5116:
5070:
4991:
4942:
4856:
4787:
4702:
4654:
4543:
4494:
4437:
4213:
3731:
3495:
3359:
3248:
3148:
3128:
3108:
3088:
3050:
3027:
2983:
2799:
2723:if any of the following conditions are satisfied:
2693:
2571:, then the limit is zero, while if the constraint
2539:
2299:
2177:
1944:
1901:
1854:
1797:
1695:
1586:
1454:
1415:
1391:
1368:
1216:
1180:
1152:
1128:
1059:values, with weights given by their probabilities
1011:
768:
687:
506:
56:
9096:
9062:
9038:
9004:
8980:
8940:
7876:: The expected value of a measurable function of
6326:whenever the right-hand side is well-defined. By
4599:
4586:
3256:respectively, have the same finite area, i.e. if
1720:are the possible outcomes of the random variable
1095:), the weighted average is given by the standard
699:), while a variety of bracket notations (such as
14086:
11302:
11258:
11186:
11118:
10837:
10595:
10569:
10266:
9709:
9645:
9530:
8428:a random variable with finite expectation. Then
7554:
6082:of expectation: The expected value operator (or
2561:does not exist: if the limits are taken so that
2188:
1855:{\displaystyle p_{i}={\tfrac {c}{i\cdot 2^{i}}}}
575:In the foreword to his treatise, Huygens wrote:
11645:), we can use this inversion formula to obtain
8136:is a function of several random variables, and
5580:{\displaystyle X\sim \mathrm {Par} (\alpha ,k)}
2863:is a Borel set with Lebesgue measure less than
13849:
13598:
13575:
13234:
13158:Choice and Chance with One Thousand Exercises.
11039:{\displaystyle \operatorname {E} <\infty .}
9253:be a random variable distributed uniformly on
2991:for any absolutely continuous random variable
13936:
13825:Continuous univariate distributions. Volume 1
13749:
13497:
13477:
13461:
13441:
13425:
13409:
13389:
13373:
13081:Bulletin of the American Mathematical Society
13078:
12369:{\displaystyle {\mathbf {1} }_{\mathcal {A}}}
4857:{\displaystyle X\sim \mathrm {Geometric} (p)}
4703:{\displaystyle X\sim \mathrm {Po} (\lambda )}
3438:
3413:
3345:
3320:
376:
13138:: CS1 maint: multiple names: authors list (
12991:. Springer texts in statistics. p. 47.
12799:
12783:
12777:
12755:
12677:
12646:
12640:
12625:
12605:
11354:
11341:
10904:
10879:
10539:
10508:
10349:
10335:
10134:
10109:
9786:
9761:
9423:
9417:
8399:and its applications to probability theory.
6122:in the sense that, for any random variables
2999:
857:Random variables with finitely many outcomes
13719:
13669:
13657:
13493:
13457:
13405:
13325:
13313:
13301:
13289:
13274:
13195:
12957:"PROBABILITY AND STATISTICS FOR ECONOMISTS"
7589:denotes the underlying probability measure.
3506:
1430:to the expected value, a fact known as the
815:. It is defined component by component, as
541:by French writer and amateur mathematician
13943:
13929:
13142:) CS1 maint: numeric names: authors list (
12425:is a discrete random variable with values
10998:{\displaystyle |X_{n}|\leq Y\leq +\infty }
9436:being the indicator function of the event
7126:, it follows that for any random variable
5939:. Note that the letters "a.s." stand for "
4360:). Hence, in this case the expectation of
3220:
3175:
615:More than a hundred years later, in 1814,
383:
369:
13357:
13160:Fifth edition. Deighton Bell, Cambridge.
12986:
12028:
11991:
11943:
11906:
11773:
11729:
11700:
11626:
11616:
11550:
11506:
11375:Relationship with characteristic function
10813:{\displaystyle \operatorname {E} \leq C.}
10372:{\displaystyle \{X_{i}\}_{i=0}^{\infty }}
9937:An example is easily obtained by setting
8096:
8066:
7480:
7440:
7335:
6068:{\displaystyle \operatorname {E} \geq 0.}
5904:
5636:
5507:
5363:
5309:
5248:{\displaystyle X\sim N(\mu ,\sigma ^{2})}
5169:
5040:
3603:
3483:
3443:
3350:
3293:
3221:
3176:
2974:
2953:
2937:
2787:
2681:
2454:
2397:
2287:
2151:
2147:
2143:
2139:
2090:
2086:
1974:
1679:
1511:
1494:
681:
500:
13713:(English translation, published in 1714)
12897:β a generalization of the expected value
12404:
12129:
10713:{\displaystyle \operatorname {E} \leq C}
10147:be a sequence of random variables, with
3058:by a nearby equality of areas. In fact,
3035:can also be defined on the graph of its
1168:after the toss. The possible values for
1106:
643:to denote "expected value" goes back to
465:The expected value of a random variable
438:. Informally, the expected value is the
13705:
13686:
12193:to justify estimating probabilities by
10545:{\displaystyle \{X_{n}\geq 0:n\geq 0\}}
10192:{\displaystyle 0\leq X_{n}\leq X_{n+1}}
7146:with well-defined expectation, one has
4495:{\displaystyle 0\cdot (1-p)+1\cdot p=p}
4372:Expected values of common distributions
4319:with remaining probability. Similarly,
3116:if and only if the two surfaces in the
3089:{\displaystyle \operatorname {E} =\mu }
2225:over that interval. The expectation of
561:
14:
14087:
13795:
13775:
13645:
13626:
13614:
13594:
13590:
13563:
13521:
13262:
13250:
13222:
13183:
13118:A philosophical essay on probabilities
10917:be a sequence of random variables. If
9317:define a sequence of random variables
8550:, one obtains the Lyapunov inequality
7122:as discussed above, together with the
6001:{\displaystyle \left\{X<0\right\}.}
2609:Arbitrary real-valued random variables
2317:absolutely continuous random variables
13924:
13538:
13534:
13532:
13530:
13510:Johnson, Kotz & Balakrishnan 1994
13352:. Technische Hochschule Brandenburg.
12859:is the same as the expected value of
12376:is the indicator function of the set
12200:The expected values of the powers of
7372:as applied to this representation of
7017:In particular, for a random variable
7010:{\displaystyle \operatorname {E} =c.}
6822:{\displaystyle \operatorname {E} =0,}
6421:{\displaystyle a_{i}(1\leq i\leq N),}
2209:. This means that the probability of
1945:{\displaystyle c={\tfrac {1}{\ln 2}}}
1021:Since the probabilities must satisfy
13881:
13602:
13481:
13465:
13445:
13429:
13413:
13393:
13377:
13304:, Theorems 31.7 and 31.8 and p. 422.
13238:
13207:
12149:based on available data gained from
9143:In general, it is not the case that
5117:{\displaystyle X\sim \exp(\lambda )}
4235:are both finite. Due to the formula
4227:exists and is finite if and only if
2814:, in which the integral is Lebesgue.
2706:is defined via weighted averages of
14095:Theory of probability distributions
13337:
13097:
13043:
10029:{\displaystyle Y_{n}=X_{n+1}-X_{n}}
9802:the expected value operator is not
7950:has a probability density function
7874:Law of the unconscious statistician
7504:valued in the nonnegative integers
7249:, as calculated in the table above.
2897:law of the unconscious statistician
621:ThΓ©orie analytique des probabilitΓ©s
532:, which seeks to divide the stakes
24:
13887:Introduction to probability models
13527:
12987:Wasserman, Larry (December 2010).
12954:
12736:
12510:
12482:
12385:
12360:
12321:
12302:
12290:
12272:
12115:{\displaystyle \operatorname {E} }
12088:
11858:
11840:{\displaystyle \operatorname {E} }
11813:
11652:
11292:
11267:
11195:
11152:
11127:
11095:
11077:
11055:
11030:
11012:
10992:
10866:(a.s.) and applying Fatou's lemma.
10786:
10682:
10604:
10559:
10461:
10456:
10417:
10390:
10364:
10300:
10275:
9900:
9895:
9856:
9829:
9795:{\displaystyle \{Y_{n}:n\geq 0\},}
9719:
9697:
9660:
9655:
9499:
9175:
9150:
9067:
9009:
8945:
8802:
8755:
8727:
8617:
8563:
8499:Part of the assertion is that the
8462:
8441:
8280:
8222:
8195:
8033:
7842:
7827:
7806:
7767:
7752:
7731:
7652:
7634:
7620:{\displaystyle \operatorname {E} }
7599:
7549:
7515:
7458:
7411:
7383:
7327:
7322:
7296:
7181:
7158:
7074:
7053:
7044:
6983:
6912:
6894:
6785:
6686:
6668:
6654:{\displaystyle \operatorname {E} }
6636:
6622:{\displaystyle \operatorname {E} }
6604:
6526:
6435:
6294:
6266:
6244:
6226:
6198:
6111:{\displaystyle \operatorname {E} }
6093:
6044:
5843:
5838:
5776:
5773:
5770:
5767:
5764:
5761:
5697:
5605:
5558:
5555:
5552:
5471:
5466:
5301:
5296:
5142:
4888:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4820:
4817:
4734:
4687:
4684:
4590:
4250:, this is the case if and only if
4198:
4173:
4165:
4140:
4118:
4093:
4085:
4060:
4050:
4037:
4012:
4004:
3979:
3969:
3956:
3931:
3923:
3898:
3869:
3844:
3818:
3588:
3554:
3461:
3406:
3378:
3313:
3271:
3065:
2941:
2929:
2906:
2840:equal to zero, the probability of
2741:
2685:
2673:
2650:
2601:for more general random variables
2315:, and the corresponding theory of
2267:
2262:
2236:
1959:
1664:
1630:
1568:
1543:
1521:
1500:
1485:
1231:
916:
776:are commonly used in physics, and
27:Average value of a random variable
25:
14111:
13046:The American Mathematical Monthly
12889:
12726:can be calculated by the formula
12220:are expected values of powers of
11379:The probability density function
10910:{\displaystyle \{X_{n}:n\geq 0\}}
10140:{\displaystyle \{X_{n}:n\geq 0\}}
9429:{\displaystyle \mathbf {1} \{A\}}
8882:, and is particularly well-known.
1762:does not have finite expectation.
905:of occurring. The expectation of
13990:
13976:cumulative distribution function
12447:along the rod and having masses
12432:and corresponding probabilities
12353:
12314:
10859:{\textstyle X=\liminf _{n}X_{n}}
9413:
9344:
7283:cumulative distribution function
7105:As a consequence of the formula
3811:, it is then natural to define:
3531:, with associated probabilities
3503:also with convergent integrals.
3037:cumulative distribution function
2819:cumulative distribution function
2547:The limit of this expression as
1902:{\displaystyle i=1,2,3,\ldots ,}
1399:times and computes the average (
784:in Russian-language literature.
89:
14063:probability-generating function
13679:
13331:
13163:
12396:{\displaystyle {\mathcal {A}}.}
12250:of this estimate gets smaller.
12246:of the sample gets larger, the
12122:also follows directly from the
10092:is as in the previous example.
8164:
7241:is given by the probability of
7037:with well-defined expectation,
3207:
3201:
2213:taking on a value in any given
2193:Now consider a random variable
1164:showing on the top face of the
769:{\displaystyle {\overline {X}}}
590:In the mid-nineteenth century,
37:"E(X)" redirects here. For the
32:Expected value (disambiguation)
13150:
13109:
13091:
13072:
13037:
13005:
12980:
12948:
12932:"Expectation | Mean | Average"
12924:
12792:
12765:
12743:
12733:
12710:
12670:
12663:
12653:
12634:
12611:{\displaystyle |\psi \rangle }
12598:
12570:
12526:
12522:
12516:
12507:
12501:
12488:
12476:
12470:
12327:
12308:
12296:
12278:
12109:
12106:
12100:
12094:
12065:
12059:
12006:
12000:
11979:
11973:
11940:
11934:
11921:
11915:
11879:
11876:
11870:
11864:
11834:
11831:
11825:
11819:
11770:
11764:
11715:
11709:
11673:
11670:
11664:
11658:
11621:
11588:
11582:
11547:
11541:
11479:
11473:
11321:
11298:
11286:
11273:
11250:: In some cases, the equality
11223:
11202:
11164:
11158:
11146:
11133:
11089:
11083:
11070:
11062:
11024:
11018:
10976:
10961:
10934:
10798:
10792:
10763:
10701:
10688:
10623:
10610:
10588:
10565:
10480:
10467:
10312:
10306:
10294:
10281:
10242:
9919:
9906:
9716:
9679:
9666:
9652:
9518:
9505:
9476:
9272:
9260:
9217:
9187:
9181:
9172:
9169:
9156:
9083:
9074:
9025:
9016:
8967:
8952:
8829:
8818:
8809:
8799:
8782:
8771:
8762:
8752:
8745:
8734:
8633:
8624:
8579:
8570:
8483:
8480:
8474:
8468:
8456:
8453:
8447:
8438:
8346:
8340:
8325:
8315:
8311:
8305:
8290:
8286:
8234:
8228:
8213:
8201:
8093:
8087:
8081:
8075:
8054:
8051:
8045:
8039:
7966:
7960:
7914:
7908:
7854:
7848:
7839:
7833:
7821:
7812:
7779:
7773:
7764:
7758:
7746:
7737:
7664:
7658:
7646:
7640:
7614:
7605:
7569:
7557:
7527:
7521:
7477:
7471:
7437:
7434:
7428:
7416:
7395:
7389:
7348:
7342:
7308:
7302:
7268:
7262:
7196:
7188:
7174:
7170:
7164:
7154:
7086:
7080:
7068:
7065:
7059:
7050:
6995:
6989:
6924:
6918:
6906:
6900:
6807:
6803:
6795:
6791:
6698:
6692:
6680:
6674:
6648:
6642:
6616:
6610:
6545:
6532:
6412:
6394:
6306:
6300:
6281:
6272:
6256:
6250:
6238:
6232:
6216:
6204:
6105:
6099:
6056:
6050:
5879:
5859:
5799:
5780:
5574:
5562:
5422:
5410:
5242:
5223:
5111:
5105:
4986:
4974:
4912:
4899:
4851:
4845:
4697:
4691:
4628:
4615:
4538:
4526:
4471:
4459:
4432:
4420:
4285:with respective probabilities
4223:According to this definition,
4192:
4179:
4159:
4146:
4112:
4099:
4079:
4066:
4031:
4018:
3998:
3985:
3950:
3937:
3917:
3904:
3888:
3875:
3863:
3850:
3830:
3824:
3566:
3560:
3480:
3474:
3433:
3427:
3390:
3384:
3340:
3334:
3290:
3284:
3231:
3225:
3198:
3192:
3077:
3071:
2971:
2965:
2918:
2912:
2784:
2778:
2759:
2747:
2662:
2656:
2394:
2388:
2284:
2278:
2248:
2242:
2229:is then given by the integral
2077:
2062:
2053:
2038:
2029:
2014:
1971:
1965:
1642:
1636:
1512:
1491:
1243:
1237:
928:
922:
458:, the expectation is given by
156:Collectively exhaustive events
13:
1:
12917:
12165:of the underlying parameter.
10870:Dominated convergence theorem
8919:both finite, it follows that
7252:Formulas in terms of CDF: If
5930:
4438:{\displaystyle X\sim ~b(1,p)}
2715:functions. A random variable
2632:, then the expected value of
2189:Random variables with density
1766:
1040:, it is natural to interpret
795:probability density functions
787:
434:) is a generalization of the
13970:probability density function
13706:Huygens, Christiaan (1657).
12174:For a different example, in
11449:{\displaystyle \varphi _{X}}
11406:of a scalar random variable
10381:monotone convergence theorem
10100:Monotone convergence theorem
7795:If the random variables are
7627:is not necessarily equal to
5428:{\displaystyle X\sim N(0,1)}
4992:{\displaystyle X\sim U(a,b)}
4544:{\displaystyle X\sim B(n,p)}
4287:6Ο, 6(2Ο), 6(3Ο), 6(4Ο), ...
4277:Suppose the random variable
2889:probability density function
2582:is taken, then the limit is
2199:probability density function
761:
721:Another popular notation is
688:{\displaystyle \mathbb {E} }
634:
601:
507:{\displaystyle \mathbb {E} }
7:
13823:; Balakrishnan, N. (1994).
13709:De ratiociniis in ludo aleæ
13691:(2nd ed.). JHU Press.
12821:
12229:moment generating functions
10668:{\displaystyle X_{n}\geq 0}
9970:{\displaystyle Y_{0}=X_{1}}
8895:, for any random variables
3761:positive and negative parts
2855:there is a positive number
2734:on the real line such that
2329:continuous random variables
1432:strong law of large numbers
1102:
861:Consider a random variable
566:De ratiociniis in ludo aleæ
10:
14116:
14052:moment-generating function
13599:Papoulis & Pillai 2002
13576:Papoulis & Pillai 2002
13235:Papoulis & Pillai 2002
12719:{\displaystyle {\hat {A}}}
12579:{\displaystyle {\hat {A}}}
12167:
11572:For the expected value of
11456:by the inversion formula:
10943:{\displaystyle X_{n}\to X}
10772:{\displaystyle X_{n}\to X}
10251:{\displaystyle X_{n}\to X}
9485:{\displaystyle X_{n}\to 0}
9226:{\displaystyle X_{n}\to X}
8170:Concentration inequalities
4340:for each positive integer
4311:for each positive integer
3369:improper Riemann integrals
2867:, then the probability of
1403:) of the results, then as
523:
36:
29:
14047:
13999:
13988:
13964:probability mass function
13959:
13953:probability distributions
13498:Casella & Berger 2001
13478:Casella & Berger 2001
13462:Casella & Berger 2001
13442:Casella & Berger 2001
13426:Casella & Berger 2001
13410:Casella & Berger 2001
13390:Casella & Berger 2001
13374:Casella & Berger 2001
12936:www.probabilitycourse.com
11360:{\displaystyle \{X_{n}\}}
8880:CauchyβSchwarz inequality
8859:for any random variables
8685:, in the special case of
7724:, then one can show that
7247:Bernoulli random variable
4378:probability distributions
442:of the possible values a
12849:Law of total expectation
11334:holds when the sequence
10739:{\displaystyle n\geq 0.}
10218:{\displaystyle n\geq 0.}
10058:{\displaystyle n\geq 1,}
9310:{\displaystyle n\geq 1,}
8388:are of course 100%. The
8184:and any positive number
7376:, it can be proved that
3507:Infinite expected values
2851:for any positive number
1423:grows, the average will
417:mathematical expectation
326:Law of total probability
321:Conditional independence
210:Exponential distribution
195:Probability distribution
14058:characteristic function
13895:10.1016/C2017-0-01324-1
13725:Probability and measure
13687:Edwards, A.W.F (2002).
13156:Whitworth, W.A. (1901)
12838:Expectation (epistemic)
12833:Conditional expectation
11428:characteristic function
9815:{\displaystyle \sigma }
8709:are numbers satisfying
6768:{\displaystyle Z\geq 0}
6587:{\displaystyle X\leq Y}
6030:{\displaystyle X\geq 0}
5962:{\displaystyle X\geq 0}
4289:. Then it follows that
2727:there is a nonnegative
1798:{\displaystyle x_{i}=i}
661:espΓ©rance mathΓ©matique.
556:In Dutch mathematician
492:also often stylized as
305:Conditional probability
13855:Pillai, S. Unnikrishna
13120:. Dover Publications.
12882:
12812:
12720:
12687:
12612:
12580:
12542:
12410:
12397:
12370:
12337:
12255:statistical estimation
12214:moments about the mean
12116:
12075:
12039:
11957:
11849:FubiniβTonelli theorem
11841:
11798:
11635:
11595:
11564:
11450:
11420:
11400:
11361:
11328:
11237:
11171:
11102:
11040:
10999:
10944:
10911:
10860:
10814:
10773:
10740:
10714:
10669:
10633:
10546:
10490:
10460:
10421:
10373:
10322:
10252:
10219:
10193:
10141:
10086:
10059:
10030:
9971:
9929:
9899:
9860:
9816:
9796:
9746:
9631:
9608:
9486:
9459:Then, it follows that
9453:
9430:
9399:
9311:
9282:
9247:
9227:
9194:
9122:
8867:. The special case of
8853:
8669:
8493:
8369:
8272:Chebyshev's inequality
8250:
8150:
8130:
8110:
8020:
8000:
7976:
7944:
7924:
7893:
7864:
7789:
7714:
7694:
7674:
7621:
7579:
7553:
7494:
7368:. As a consequence of
7358:
7275:
7207:
7140:
7096:
7031:
7011:
6963:
6934:
6878:
6849:
6823:
6769:
6743:
6742:{\displaystyle Z=Y-X,}
6708:
6655:
6623:
6588:
6555:
6515:
6466:
6422:
6371:
6344:
6320:
6182:
6159:
6136:
6112:
6069:
6031:
6002:
5963:
5915:
5806:
5733:
5581:
5524:
5429:
5380:
5249:
5193:
5118:
5072:
4993:
4944:
4892:
4858:
4789:
4738:
4704:
4656:
4579:
4545:
4496:
4439:
4215:
3733:
3592:
3517:St. Petersburg paradox
3497:
3361:
3250:
3150:
3130:
3110:
3090:
3052:
3029:
3012:
2985:
2801:
2695:
2541:
2301:
2179:
1946:
1903:
1856:
1799:
1746:Riemann series theorem
1697:
1668:
1621:countably infinite set
1588:
1456:
1417:
1393:
1370:
1218:
1182:
1160:will be the number of
1154:
1130:
1112:
1013:
770:
689:
639:The use of the letter
632:
613:
588:
508:
247:Continuous or discrete
200:Bernoulli distribution
58:
13874:(Erratum:
13759:Statistical inference
13545:mathworld.wolfram.com
12895:Nonlinear expectation
12883:
12813:
12721:
12688:
12613:
12581:
12543:
12408:
12398:
12371:
12338:
12130:Uses and applications
12117:
12076:
12074:{\displaystyle g(x).}
12040:
11958:
11842:
11799:
11636:
11596:
11565:
11451:
11421:
11401:
11399:{\displaystyle f_{X}}
11369:uniformly integrable.
11362:
11329:
11248:Uniform integrability
11238:
11172:
11103:
11041:
11000:
10945:
10912:
10861:
10823:is by observing that
10815:
10774:
10741:
10715:
10670:
10634:
10547:
10491:
10440:
10401:
10374:
10323:
10253:
10220:
10194:
10142:
10087:
10085:{\displaystyle X_{n}}
10060:
10031:
9972:
9930:
9879:
9840:
9817:
9797:
9747:
9632:
9609:
9487:
9454:
9431:
9400:
9312:
9283:
9248:
9228:
9195:
9123:
8854:
8670:
8540:for positive numbers
8494:
8397:mathematical analysis
8390:Kolmogorov inequality
8370:
8251:
8151:
8131:
8111:
8021:
8001:
7977:
7975:{\displaystyle f(x),}
7945:
7925:
7923:{\displaystyle g(X),}
7894:
7865:
7790:
7715:
7695:
7675:
7622:
7580:
7533:
7495:
7359:
7285:of a random variable
7276:
7208:
7141:
7097:
7032:
7012:
6972:for some real number
6964:
6935:
6879:
6850:
6824:
6770:
6744:
6709:
6656:
6624:
6589:
6565:on this vector space.
6556:
6495:
6446:
6423:
6372:
6370:{\displaystyle X_{i}}
6345:
6321:
6183:
6160:
6137:
6113:
6070:
6032:
6003:
5964:
5916:
5807:
5734:
5582:
5525:
5430:
5381:
5250:
5194:
5119:
5073:
4994:
4945:
4872:
4859:
4790:
4718:
4705:
4657:
4559:
4546:
4497:
4440:
4216:
3734:
3572:
3498:
3362:
3251:
3156:-plane, described by
3151:
3131:
3111:
3091:
3053:
3030:
3003:
2986:
2848:is also equal to zero
2836:of real numbers with
2827:absolutely continuous
2802:
2721:absolutely continuous
2696:
2542:
2302:
2180:
1947:
1904:
1857:
1800:
1750:mathematical analysis
1698:
1648:
1589:
1457:
1418:
1394:
1371:
1219:
1204:. The expectation of
1183:
1155:
1140:. More specifically,
1131:
1110:
1014:
771:
690:
625:
619:published his tract "
608:
577:
570:theory of probability
509:
205:Binomial distribution
59:
57:{\displaystyle e^{x}}
14100:Gambling terminology
13851:Papoulis, Athanasios
13721:Billingsley, Patrick
13338:Uhl, Roland (2023).
13098:Huygens, Christian.
12872:
12730:
12701:
12622:
12594:
12561:
12461:
12380:
12347:
12269:
12191:law of large numbers
12085:
12053:
11967:
11855:
11810:
11649:
11605:
11594:{\displaystyle g(X)}
11576:
11460:
11433:
11410:
11383:
11338:
11254:
11182:
11114:
11052:
11009:
10957:
10921:
10876:
10827:
10783:
10750:
10724:
10679:
10646:
10556:
10505:
10387:
10332:
10262:
10229:
10203:
10151:
10106:
10069:
10040:
9981:
9941:
9826:
9806:
9758:
9641:
9618:
9496:
9463:
9440:
9409:
9321:
9292:
9257:
9237:
9204:
9147:
8935:
8886:Minkowski inequality
8724:
8554:
8432:
8277:
8192:
8140:
8120:
8030:
8010:
7990:
7954:
7934:
7902:
7880:
7803:
7728:
7704:
7684:
7631:
7596:
7512:
7380:
7370:integration by parts
7293:
7274:{\displaystyle F(x)}
7256:
7150:
7130:
7041:
7021:
6980:
6947:
6891:
6862:
6833:
6782:
6753:
6718:
6665:
6633:
6601:
6572:
6432:
6381:
6354:
6334:
6191:
6169:
6146:
6126:
6090:
6084:expectation operator
6041:
6015:
5973:
5947:
5817:
5751:
5592:
5542:
5440:
5398:
5260:
5211:
5129:
5090:
5004:
4962:
4869:
4807:
4715:
4674:
4556:
4514:
4450:
4405:
3815:
3551:
3375:
3260:
3160:
3140:
3120:
3109:{\displaystyle \mu }
3100:
3062:
3042:
3019:
2903:
2738:
2647:
2640:, is defined as the
2591:converges absolutely
2364:
2321:piecewise continuous
2313:Lebesgue integration
2233:
2201:given by a function
1956:
1913:
1866:
1809:
1776:
1758:converges absolutely
1627:
1482:
1446:
1407:
1383:
1228:
1208:
1172:
1144:
1120:
913:
803:Lebesgue integration
753:
677:
659:, and in French for
657:esperanza matemΓ‘tica
617:Pierre-Simon Laplace
496:
469:is often denoted by
460:Lebesgue integration
413:expectation operator
331:Law of large numbers
300:Marginal probability
225:Poisson distribution
74:Part of a series on
66:Exponential function
41:
13660:, pp. 81, 277.
13541:"Expectation Value"
13539:Weisstein, Eric W.
13359:10.25933/opus4-2986
12415:classical mechanics
12263:Monte Carlo methods
12081:The expression for
10368:
8931:is also finite and
8888:: given any number
8693:HΓΆlder's inequality
8404:Jensen's inequality
8385:standard deviations
8174:Markov's inequality
7467:
7415:
7331:
7124:triangle inequality
6962:{\displaystyle X=c}
6877:{\displaystyle X=Y}
6848:{\displaystyle X=0}
6778:Non-degeneracy: If
6011:Non-negativity: If
5847:
5609:
5475:
5305:
5146:
5021:
3470:
3410:
3317:
3280:
3096:with a real number
2729:measurable function
2421:
2381:
2340:Cauchy distribution
2325:Riemann integration
2271:
558:Christiaan Huygens'
290:Complementary event
232:Probability measure
220:Pareto distribution
215:Normal distribution
14024:standard deviation
13817:Johnson, Norman L.
13021:10.1002/0471725161
12878:
12808:
12716:
12683:
12608:
12576:
12538:
12411:
12393:
12366:
12333:
12187:indicator function
12141:, where one seeks
12124:Plancherel theorem
12112:
12071:
12035:
11953:
11837:
11794:
11631:
11591:
11560:
11446:
11426:is related to its
11416:
11396:
11357:
11324:
11310:
11266:
11233:
11194:
11167:
11126:
11098:
11036:
10995:
10940:
10907:
10856:
10845:
10810:
10769:
10736:
10710:
10665:
10629:
10603:
10577:
10542:
10486:
10369:
10348:
10318:
10274:
10248:
10215:
10189:
10137:
10082:
10055:
10026:
9967:
9925:
9812:
9792:
9742:
9723:
9659:
9630:{\displaystyle n.}
9627:
9604:
9596:
9565:
9482:
9452:{\displaystyle A.}
9449:
9426:
9395:
9380:
9307:
9278:
9243:
9223:
9190:
9118:
8849:
8687:probability spaces
8665:
8489:
8365:
8246:
8146:
8126:
8106:
8016:
7996:
7972:
7940:
7920:
7892:{\displaystyle X,}
7889:
7860:
7785:
7710:
7690:
7670:
7617:
7575:
7490:
7450:
7401:
7366:Lebesgue-Stieltjes
7354:
7314:
7271:
7228:indicator function
7203:
7136:
7092:
7027:
7007:
6959:
6930:
6874:
6845:
6819:
6765:
6739:
6704:
6651:
6619:
6584:
6551:
6418:
6367:
6340:
6316:
6314:
6181:{\displaystyle a,}
6178:
6158:{\displaystyle Y,}
6155:
6132:
6108:
6065:
6027:
5998:
5959:
5911:
5830:
5802:
5729:
5724:
5595:
5577:
5520:
5458:
5425:
5376:
5288:
5245:
5189:
5132:
5114:
5068:
5007:
4989:
4940:
4854:
4785:
4700:
4652:
4541:
4492:
4435:
4315:, and takes value
4211:
4206:
3759:, one defines the
3729:
3493:
3453:
3396:
3357:
3303:
3263:
3246:
3146:
3126:
3106:
3086:
3048:
3025:
3013:
2981:
2899:, it follows that
2832:for any Borel set
2797:
2691:
2537:
2407:
2367:
2297:
2254:
2175:
2170:
2131:
2116:
2101:
2075:
2051:
2027:
1987:
1942:
1940:
1899:
1852:
1850:
1795:
1693:
1584:
1452:
1413:
1389:
1366:
1214:
1178:
1150:
1126:
1113:
1009:
766:
685:
629:mathematical hope.
504:
397:probability theory
341:Boole's inequality
277:Stochastic process
166:Mutual exclusivity
83:Probability theory
54:
14082:
14081:
13982:quantile function
13904:978-0-12-814346-9
13464:, Example 2.2.2;
13392:, Example 2.2.3;
12881:{\displaystyle m}
12795:
12768:
12713:
12666:
12637:
12573:
12552:quantum mechanics
12170:Estimation theory
12047:Fourier transform
11898:
11692:
11498:
11419:{\displaystyle X}
11301:
11257:
11185:
11117:
10836:
10594:
10568:
10265:
10225:Furthermore, let
9708:
9644:
9595:
9564:
9379:
9281:{\displaystyle .}
9246:{\displaystyle U}
8360:
8303:
8241:
8188:, it states that
8149:{\displaystyle f}
8129:{\displaystyle g}
8019:{\displaystyle g}
7999:{\displaystyle f}
7943:{\displaystyle X}
7799:, then generally
7713:{\displaystyle Y}
7693:{\displaystyle X}
7506:{0, 1, 2, 3, ...}
7139:{\displaystyle X}
7030:{\displaystyle X}
6568:Monotonicity: If
6350:random variables
6343:{\displaystyle N}
6135:{\displaystyle X}
5937:Lebesgue integral
5928:
5927:
5902:
5828:
5705:
5682:
5675:
5456:
5455:
5349:
5326:
5286:
5285:
5187:
5066:
5038:
4938:
4777:
4597:
4416:
4332:with probability
4303:with probability
4171:
4138:
4131:
4091:
4058:
4010:
3977:
3929:
3896:
3688:
3669:
3650:
3631:
3205:
3149:{\displaystyle y}
3129:{\displaystyle x}
3051:{\displaystyle F}
3028:{\displaystyle X}
2642:Lebesgue integral
2627:probability space
2621:is a real-valued
2617:. In general, if
2532:
2472:
2452:
2169:
2130:
2115:
2100:
2074:
2050:
2026:
1978:
1939:
1849:
1579:
1560:
1538:
1509:
1498:
1455:{\displaystyle X}
1416:{\displaystyle n}
1392:{\displaystyle n}
1376:If one rolls the
1358:
1339:
1320:
1301:
1282:
1263:
1217:{\displaystyle X}
1181:{\displaystyle X}
1153:{\displaystyle X}
1129:{\displaystyle X}
764:
655:, in Spanish for
592:Pafnuty Chebyshev
543:Chevalier de MΓ©rΓ©
530:problem of points
425:expectation value
393:
392:
295:Joint probability
242:Bernoulli process
141:Probability space
18:Expectation value
16:(Redirected from
14107:
13994:
13945:
13938:
13931:
13922:
13921:
13916:
13883:Ross, Sheldon M.
13878:
13872:
13846:
13812:
13792:
13772:
13755:Berger, Roger L.
13746:
13716:
13714:
13702:
13673:
13670:Billingsley 1995
13667:
13661:
13658:Billingsley 1995
13655:
13649:
13643:
13630:
13624:
13618:
13612:
13606:
13593:, Section IX.6;
13588:
13579:
13573:
13567:
13561:
13555:
13554:
13552:
13551:
13536:
13525:
13519:
13513:
13507:
13501:
13496:, Example 21.1;
13494:Billingsley 1995
13491:
13485:
13475:
13469:
13460:, Example 21.3;
13458:Billingsley 1995
13455:
13449:
13439:
13433:
13423:
13417:
13408:, Example 21.4;
13406:Billingsley 1995
13403:
13397:
13387:
13381:
13371:
13365:
13363:
13361:
13351:
13335:
13329:
13328:, Theorem 16.11.
13326:Billingsley 1995
13323:
13317:
13316:, Theorem 16.13.
13314:Billingsley 1995
13311:
13305:
13302:Billingsley 1995
13299:
13293:
13290:Billingsley 1995
13287:
13278:
13275:Billingsley 1995
13272:
13266:
13260:
13254:
13248:
13242:
13241:, Section 2.4.2.
13232:
13226:
13220:
13211:
13210:, Section 2.4.1.
13205:
13199:
13196:Billingsley 1995
13193:
13187:
13181:
13175:
13174:
13167:
13161:
13154:
13148:
13147:
13137:
13129:
13113:
13107:
13106:
13104:
13095:
13089:
13088:
13076:
13070:
13069:
13041:
13035:
13034:
13009:
13003:
13002:
12984:
12978:
12977:
12975:
12974:
12968:
12962:. Archived from
12961:
12952:
12946:
12945:
12943:
12942:
12928:
12887:
12885:
12884:
12879:
12828:Central tendency
12817:
12815:
12814:
12809:
12807:
12806:
12797:
12796:
12788:
12776:
12775:
12770:
12769:
12761:
12751:
12750:
12725:
12723:
12722:
12717:
12715:
12714:
12706:
12692:
12690:
12689:
12684:
12673:
12668:
12667:
12659:
12656:
12639:
12638:
12630:
12617:
12615:
12614:
12609:
12601:
12585:
12583:
12582:
12577:
12575:
12574:
12566:
12547:
12545:
12544:
12539:
12534:
12533:
12500:
12499:
12402:
12400:
12399:
12394:
12389:
12388:
12375:
12373:
12372:
12367:
12365:
12364:
12363:
12357:
12356:
12342:
12340:
12339:
12334:
12326:
12325:
12324:
12318:
12317:
12295:
12294:
12293:
12259:machine learning
12226:
12180:utility function
12121:
12119:
12118:
12113:
12080:
12078:
12077:
12072:
12044:
12042:
12041:
12036:
12027:
12026:
11996:
11995:
11994:
11962:
11960:
11959:
11954:
11933:
11932:
11911:
11910:
11909:
11899:
11897:
11886:
11846:
11844:
11843:
11838:
11803:
11801:
11800:
11795:
11784:
11780:
11763:
11762:
11753:
11752:
11734:
11733:
11732:
11705:
11704:
11703:
11693:
11691:
11680:
11640:
11638:
11637:
11632:
11630:
11629:
11620:
11619:
11600:
11598:
11597:
11592:
11569:
11567:
11566:
11561:
11540:
11539:
11530:
11529:
11511:
11510:
11509:
11499:
11497:
11486:
11472:
11471:
11455:
11453:
11452:
11447:
11445:
11444:
11425:
11423:
11422:
11417:
11405:
11403:
11402:
11397:
11395:
11394:
11366:
11364:
11363:
11358:
11353:
11352:
11333:
11331:
11330:
11325:
11320:
11319:
11309:
11285:
11284:
11265:
11242:
11240:
11239:
11234:
11226:
11215:
11214:
11205:
11193:
11176:
11174:
11173:
11168:
11145:
11144:
11125:
11107:
11105:
11104:
11099:
11073:
11065:
11045:
11043:
11042:
11037:
11004:
11002:
11001:
10996:
10979:
10974:
10973:
10964:
10949:
10947:
10946:
10941:
10933:
10932:
10916:
10914:
10913:
10908:
10891:
10890:
10865:
10863:
10862:
10857:
10855:
10854:
10844:
10819:
10817:
10816:
10811:
10778:
10776:
10775:
10770:
10762:
10761:
10745:
10743:
10742:
10737:
10719:
10717:
10716:
10711:
10700:
10699:
10674:
10672:
10671:
10666:
10658:
10657:
10638:
10636:
10635:
10630:
10622:
10621:
10602:
10587:
10586:
10576:
10551:
10549:
10548:
10543:
10520:
10519:
10495:
10493:
10492:
10487:
10479:
10478:
10459:
10454:
10436:
10432:
10431:
10430:
10420:
10415:
10378:
10376:
10375:
10370:
10367:
10362:
10347:
10346:
10327:
10325:
10324:
10319:
10293:
10292:
10273:
10257:
10255:
10254:
10249:
10241:
10240:
10224:
10222:
10221:
10216:
10198:
10196:
10195:
10190:
10188:
10187:
10169:
10168:
10146:
10144:
10143:
10138:
10121:
10120:
10091:
10089:
10088:
10083:
10081:
10080:
10064:
10062:
10061:
10056:
10035:
10033:
10032:
10027:
10025:
10024:
10012:
10011:
9993:
9992:
9976:
9974:
9973:
9968:
9966:
9965:
9953:
9952:
9934:
9932:
9931:
9926:
9918:
9917:
9898:
9893:
9875:
9871:
9870:
9869:
9859:
9854:
9822:-additive, i.e.
9821:
9819:
9818:
9813:
9801:
9799:
9798:
9793:
9773:
9772:
9751:
9749:
9748:
9743:
9738:
9734:
9733:
9732:
9722:
9678:
9677:
9658:
9636:
9634:
9633:
9628:
9613:
9611:
9610:
9605:
9597:
9588:
9576:
9572:
9571:
9567:
9566:
9557:
9517:
9516:
9492:pointwise. But,
9491:
9489:
9488:
9483:
9475:
9474:
9458:
9456:
9455:
9450:
9435:
9433:
9432:
9427:
9416:
9404:
9402:
9401:
9396:
9391:
9387:
9386:
9382:
9381:
9372:
9347:
9333:
9332:
9316:
9314:
9313:
9308:
9287:
9285:
9284:
9279:
9252:
9250:
9249:
9244:
9232:
9230:
9229:
9224:
9216:
9215:
9199:
9197:
9196:
9191:
9168:
9167:
9127:
9125:
9124:
9119:
9114:
9113:
9109:
9100:
9099:
9092:
9091:
9086:
9077:
9066:
9065:
9056:
9055:
9051:
9042:
9041:
9034:
9033:
9028:
9019:
9008:
9007:
8998:
8997:
8993:
8984:
8983:
8976:
8975:
8970:
8955:
8944:
8943:
8930:
8918:
8910:
8902:
8898:
8894:
8877:
8866:
8862:
8858:
8856:
8855:
8850:
8845:
8844:
8840:
8827:
8826:
8821:
8812:
8798:
8797:
8793:
8780:
8779:
8774:
8765:
8748:
8737:
8719:
8708:
8701:
8683:
8678:
8674:
8672:
8671:
8666:
8661:
8660:
8656:
8647:
8643:
8642:
8641:
8636:
8627:
8607:
8606:
8602:
8593:
8589:
8588:
8587:
8582:
8573:
8549:
8539:
8537:
8517:
8513:
8498:
8496:
8495:
8490:
8427:
8419:
8378:
8374:
8372:
8371:
8366:
8361:
8359:
8358:
8349:
8332:
8318:
8304:
8301:
8293:
8269:
8261:
8255:
8253:
8252:
8247:
8242:
8237:
8220:
8187:
8183:
8180:random variable
8155:
8153:
8152:
8147:
8135:
8133:
8132:
8127:
8115:
8113:
8112:
8107:
8071:
8070:
8069:
8025:
8023:
8022:
8017:
8005:
8003:
8002:
7997:
7982:is given by the
7981:
7979:
7978:
7973:
7949:
7947:
7946:
7941:
7929:
7927:
7926:
7921:
7898:
7896:
7895:
7890:
7869:
7867:
7866:
7861:
7794:
7792:
7791:
7786:
7719:
7717:
7716:
7711:
7699:
7697:
7696:
7691:
7679:
7677:
7676:
7671:
7626:
7624:
7623:
7618:
7588:
7584:
7582:
7581:
7576:
7552:
7547:
7507:
7503:
7499:
7497:
7496:
7491:
7466:
7461:
7414:
7409:
7375:
7363:
7361:
7360:
7355:
7330:
7325:
7288:
7280:
7278:
7277:
7272:
7244:
7240:
7236:
7225:
7212:
7210:
7209:
7204:
7199:
7191:
7177:
7157:
7145:
7143:
7142:
7137:
7121:
7112:
7101:
7099:
7098:
7093:
7036:
7034:
7033:
7028:
7016:
7014:
7013:
7008:
6975:
6968:
6966:
6965:
6960:
6939:
6937:
6936:
6931:
6883:
6881:
6880:
6875:
6854:
6852:
6851:
6846:
6828:
6826:
6825:
6820:
6806:
6798:
6774:
6772:
6771:
6766:
6748:
6746:
6745:
6740:
6713:
6711:
6710:
6705:
6660:
6658:
6657:
6652:
6628:
6626:
6625:
6620:
6593:
6591:
6590:
6585:
6560:
6558:
6557:
6552:
6544:
6543:
6525:
6524:
6514:
6509:
6491:
6487:
6486:
6485:
6476:
6475:
6465:
6460:
6427:
6425:
6424:
6419:
6393:
6392:
6376:
6374:
6373:
6368:
6366:
6365:
6349:
6347:
6346:
6341:
6325:
6323:
6322:
6317:
6315:
6187:
6185:
6184:
6179:
6164:
6162:
6161:
6156:
6141:
6139:
6138:
6133:
6117:
6115:
6114:
6109:
6081:
6080:
6074:
6072:
6071:
6066:
6036:
6034:
6033:
6028:
6007:
6005:
6004:
5999:
5994:
5990:
5968:
5966:
5965:
5960:
5920:
5918:
5917:
5912:
5903:
5901:
5900:
5899:
5887:
5886:
5877:
5876:
5857:
5849:
5846:
5841:
5829:
5821:
5811:
5809:
5808:
5803:
5792:
5791:
5779:
5738:
5736:
5735:
5730:
5728:
5727:
5706:
5703:
5683:
5680:
5676:
5674:
5663:
5655:
5635:
5634:
5622:
5621:
5608:
5603:
5586:
5584:
5583:
5578:
5561:
5529:
5527:
5526:
5521:
5506:
5505:
5501:
5496:
5495:
5474:
5469:
5457:
5448:
5444:
5434:
5432:
5431:
5426:
5385:
5383:
5382:
5377:
5362:
5361:
5360:
5359:
5354:
5350:
5345:
5334:
5327:
5319:
5304:
5299:
5287:
5284:
5283:
5268:
5264:
5254:
5252:
5251:
5246:
5241:
5240:
5198:
5196:
5195:
5190:
5188:
5180:
5168:
5167:
5145:
5140:
5123:
5121:
5120:
5115:
5077:
5075:
5074:
5069:
5067:
5062:
5051:
5039:
5037:
5023:
5020:
5015:
4998:
4996:
4995:
4990:
4949:
4947:
4946:
4941:
4939:
4931:
4926:
4925:
4891:
4886:
4863:
4861:
4860:
4855:
4844:
4794:
4792:
4791:
4786:
4778:
4776:
4768:
4767:
4766:
4757:
4756:
4740:
4737:
4732:
4709:
4707:
4706:
4701:
4690:
4661:
4659:
4658:
4653:
4642:
4641:
4614:
4613:
4604:
4603:
4602:
4589:
4578:
4573:
4550:
4548:
4547:
4542:
4501:
4499:
4498:
4493:
4444:
4442:
4441:
4436:
4414:
4383:
4382:
4363:
4355:
4351:
4347:
4344:and takes value
4343:
4339:
4331:
4324:
4318:
4314:
4310:
4302:
4294:
4288:
4284:
4283:1, β2,3, β4, ...
4280:
4273:
4269:
4257:
4249:
4234:
4230:
4226:
4220:
4218:
4217:
4212:
4210:
4209:
4191:
4190:
4172:
4169:
4158:
4157:
4139:
4136:
4132:
4129:
4111:
4110:
4092:
4089:
4078:
4077:
4059:
4056:
4030:
4029:
4011:
4008:
3997:
3996:
3978:
3975:
3949:
3948:
3930:
3927:
3916:
3915:
3897:
3894:
3887:
3886:
3862:
3861:
3810:
3806:
3802:
3798:
3784:
3773:
3758:
3754:
3742:
3738:
3736:
3735:
3730:
3689:
3681:
3670:
3662:
3651:
3643:
3632:
3624:
3613:
3612:
3602:
3601:
3591:
3586:
3546:
3542:
3530:
3514:
3502:
3500:
3499:
3494:
3469:
3464:
3442:
3441:
3417:
3416:
3409:
3404:
3366:
3364:
3363:
3358:
3349:
3348:
3324:
3323:
3316:
3311:
3279:
3274:
3255:
3253:
3252:
3247:
3206:
3203:
3155:
3153:
3152:
3147:
3135:
3133:
3132:
3127:
3115:
3113:
3112:
3107:
3095:
3093:
3092:
3087:
3057:
3055:
3054:
3049:
3034:
3032:
3031:
3026:
3011:
3007:
2994:
2990:
2988:
2987:
2982:
2958:
2957:
2956:
2933:
2932:
2894:
2886:
2878:
2874:
2871:being valued in
2870:
2866:
2862:
2858:
2854:
2847:
2844:being valued in
2843:
2838:Lebesgue measure
2835:
2824:
2813:
2806:
2804:
2803:
2798:
2774:
2773:
2733:
2718:
2713:
2705:
2700:
2698:
2697:
2692:
2677:
2676:
2639:
2635:
2631:
2620:
2604:
2600:
2596:
2585:
2581:
2570:
2560:
2553:
2546:
2544:
2543:
2538:
2533:
2531:
2530:
2529:
2517:
2516:
2506:
2505:
2504:
2492:
2491:
2481:
2473:
2465:
2453:
2451:
2450:
2449:
2437:
2436:
2423:
2420:
2415:
2380:
2375:
2359:
2344:
2338:is given by the
2337:
2306:
2304:
2303:
2298:
2270:
2265:
2228:
2224:
2217:is given by the
2212:
2207:real number line
2204:
2196:
2184:
2182:
2181:
2176:
2171:
2168:
2154:
2132:
2123:
2117:
2108:
2102:
2093:
2076:
2067:
2052:
2043:
2028:
2019:
2007:
2006:
1997:
1996:
1986:
1951:
1949:
1948:
1943:
1941:
1938:
1924:
1908:
1906:
1905:
1900:
1861:
1859:
1858:
1853:
1851:
1848:
1847:
1846:
1827:
1821:
1820:
1804:
1802:
1801:
1796:
1788:
1787:
1740:
1723:
1719:
1702:
1700:
1699:
1694:
1689:
1688:
1678:
1677:
1667:
1662:
1609:
1607:
1606:
1603:
1600:
1593:
1591:
1590:
1585:
1580:
1572:
1561:
1553:
1539:
1531:
1510:
1507:
1499:
1496:
1477:
1475:
1474:
1471:
1468:
1461:
1459:
1458:
1453:
1422:
1420:
1419:
1414:
1398:
1396:
1395:
1390:
1375:
1373:
1372:
1367:
1359:
1351:
1340:
1332:
1321:
1313:
1302:
1294:
1283:
1275:
1264:
1256:
1223:
1221:
1220:
1215:
1203:
1201:
1200:
1197:
1194:
1187:
1185:
1184:
1179:
1159:
1157:
1156:
1151:
1135:
1133:
1132:
1127:
1094:
1069:
1058:
1046:weighted average
1043:
1039:
1018:
1016:
1015:
1010:
1005:
1004:
995:
994:
976:
975:
966:
965:
953:
952:
943:
942:
908:
904:
886:
864:
852:
842:
832:with components
831:
824:
814:
783:
775:
773:
772:
767:
765:
757:
748:
737:
729:
718:) are all used.
717:
710:
706:
694:
692:
691:
686:
684:
672:
668:
650:
642:
596:random variables
586:
551:Pierre de Fermat
519:
513:
511:
510:
505:
503:
491:
487:
480:
476:
468:
436:weighted average
385:
378:
371:
161:Elementary event
93:
71:
70:
63:
61:
60:
55:
53:
52:
21:
14115:
14114:
14110:
14109:
14108:
14106:
14105:
14104:
14085:
14084:
14083:
14078:
14043:
13995:
13986:
13955:
13949:
13919:
13905:
13873:
13869:
13835:
13797:Feller, William
13777:Feller, William
13769:
13751:Casella, George
13735:
13712:
13699:
13682:
13677:
13676:
13668:
13664:
13656:
13652:
13644:
13633:
13629:, Section IX.7.
13625:
13621:
13617:, Section IX.6.
13613:
13609:
13601:, Section 5-4;
13597:, Section V.7;
13589:
13582:
13574:
13570:
13562:
13558:
13549:
13547:
13537:
13528:
13524:, Section II.4.
13520:
13516:
13508:
13504:
13492:
13488:
13484:, Example 2.22.
13480:, p. 103;
13476:
13472:
13468:, Example 2.21.
13456:
13452:
13448:, Example 2.20.
13440:
13436:
13432:, Example 2.18.
13424:
13420:
13416:, Example 2.19.
13404:
13400:
13396:, Example 2.17.
13388:
13384:
13380:, Example 2.16.
13372:
13368:
13349:
13336:
13332:
13324:
13320:
13312:
13308:
13300:
13296:
13288:
13281:
13273:
13269:
13261:
13257:
13249:
13245:
13237:, Section 5-3;
13233:
13229:
13225:, Section IX.2.
13221:
13214:
13206:
13202:
13194:
13190:
13182:
13178:
13169:
13168:
13164:
13155:
13151:
13131:
13130:
13114:
13110:
13102:
13096:
13092:
13077:
13073:
13058:10.2307/2309286
13042:
13038:
13031:
13011:
13010:
13006:
12999:
12985:
12981:
12972:
12970:
12966:
12959:
12955:Hansen, Bruce.
12953:
12949:
12940:
12938:
12930:
12929:
12925:
12920:
12911:Wald's equation
12906:Predicted value
12901:Population mean
12873:
12870:
12869:
12868:β indicated by
12824:
12802:
12798:
12787:
12786:
12771:
12760:
12759:
12758:
12746:
12742:
12731:
12728:
12727:
12705:
12704:
12702:
12699:
12698:
12669:
12658:
12657:
12652:
12629:
12628:
12623:
12620:
12619:
12597:
12595:
12592:
12591:
12586:operating on a
12565:
12564:
12562:
12559:
12558:
12529:
12525:
12495:
12491:
12462:
12459:
12458:
12452:
12445:
12437:
12430:
12384:
12383:
12381:
12378:
12377:
12359:
12358:
12352:
12351:
12350:
12348:
12345:
12344:
12320:
12319:
12313:
12312:
12311:
12289:
12288:
12281:
12270:
12267:
12266:
12236:arithmetic mean
12221:
12204:are called the
12176:decision theory
12172:
12132:
12086:
12083:
12082:
12054:
12051:
12050:
12013:
12009:
11990:
11989:
11985:
11968:
11965:
11964:
11928:
11924:
11905:
11904:
11900:
11890:
11885:
11856:
11853:
11852:
11811:
11808:
11807:
11758:
11754:
11739:
11735:
11728:
11727:
11723:
11722:
11718:
11699:
11698:
11694:
11684:
11679:
11650:
11647:
11646:
11625:
11624:
11615:
11614:
11606:
11603:
11602:
11577:
11574:
11573:
11535:
11531:
11516:
11512:
11505:
11504:
11500:
11490:
11485:
11467:
11463:
11461:
11458:
11457:
11440:
11436:
11434:
11431:
11430:
11411:
11408:
11407:
11390:
11386:
11384:
11381:
11380:
11377:
11348:
11344:
11339:
11336:
11335:
11315:
11311:
11305:
11280:
11276:
11261:
11255:
11252:
11251:
11222:
11210:
11206:
11201:
11189:
11183:
11180:
11179:
11140:
11136:
11121:
11115:
11112:
11111:
11069:
11061:
11053:
11050:
11049:
11010:
11007:
11006:
10975:
10969:
10965:
10960:
10958:
10955:
10954:
10928:
10924:
10922:
10919:
10918:
10886:
10882:
10877:
10874:
10873:
10850:
10846:
10840:
10828:
10825:
10824:
10784:
10781:
10780:
10757:
10753:
10751:
10748:
10747:
10725:
10722:
10721:
10695:
10691:
10680:
10677:
10676:
10653:
10649:
10647:
10644:
10643:
10617:
10613:
10598:
10582:
10578:
10572:
10557:
10554:
10553:
10515:
10511:
10506:
10503:
10502:
10474:
10470:
10455:
10444:
10426:
10422:
10416:
10405:
10400:
10396:
10388:
10385:
10384:
10363:
10352:
10342:
10338:
10333:
10330:
10329:
10288:
10284:
10269:
10263:
10260:
10259:
10236:
10232:
10230:
10227:
10226:
10204:
10201:
10200:
10199:(a.s) for each
10177:
10173:
10164:
10160:
10152:
10149:
10148:
10116:
10112:
10107:
10104:
10103:
10076:
10072:
10070:
10067:
10066:
10041:
10038:
10037:
10020:
10016:
10001:
9997:
9988:
9984:
9982:
9979:
9978:
9961:
9957:
9948:
9944:
9942:
9939:
9938:
9913:
9909:
9894:
9883:
9865:
9861:
9855:
9844:
9839:
9835:
9827:
9824:
9823:
9807:
9804:
9803:
9768:
9764:
9759:
9756:
9755:
9728:
9724:
9712:
9707:
9703:
9673:
9669:
9648:
9642:
9639:
9638:
9619:
9616:
9615:
9586:
9555:
9548:
9544:
9537:
9533:
9512:
9508:
9497:
9494:
9493:
9470:
9466:
9464:
9461:
9460:
9441:
9438:
9437:
9412:
9410:
9407:
9406:
9370:
9363:
9359:
9352:
9348:
9343:
9328:
9324:
9322:
9319:
9318:
9293:
9290:
9289:
9258:
9255:
9254:
9238:
9235:
9234:
9211:
9207:
9205:
9202:
9201:
9163:
9159:
9148:
9145:
9144:
9141:
9105:
9101:
9095:
9094:
9093:
9087:
9082:
9081:
9073:
9061:
9060:
9047:
9043:
9037:
9036:
9035:
9029:
9024:
9023:
9015:
9003:
9002:
8989:
8985:
8979:
8978:
8977:
8971:
8966:
8965:
8951:
8939:
8938:
8936:
8933:
8932:
8920:
8912:
8904:
8900:
8896:
8889:
8868:
8864:
8860:
8836:
8832:
8828:
8822:
8817:
8816:
8808:
8789:
8785:
8781:
8775:
8770:
8769:
8761:
8744:
8733:
8725:
8722:
8721:
8710:
8703:
8696:
8681:
8676:
8652:
8648:
8637:
8632:
8631:
8623:
8616:
8612:
8611:
8598:
8594:
8583:
8578:
8577:
8569:
8562:
8558:
8557:
8555:
8552:
8551:
8541:
8533:
8524:
8515:
8504:
8433:
8430:
8429:
8425:
8422:convex function
8407:
8376:
8354:
8350:
8333:
8331:
8314:
8300:
8289:
8278:
8275:
8274:
8263:
8259:
8221:
8219:
8193:
8190:
8189:
8185:
8181:
8167:
8141:
8138:
8137:
8121:
8118:
8117:
8065:
8064:
8060:
8031:
8028:
8027:
8011:
8008:
8007:
7991:
7988:
7987:
7955:
7952:
7951:
7935:
7932:
7931:
7903:
7900:
7899:
7881:
7878:
7877:
7804:
7801:
7800:
7729:
7726:
7725:
7705:
7702:
7701:
7685:
7682:
7681:
7632:
7629:
7628:
7597:
7594:
7593:
7586:
7548:
7537:
7513:
7510:
7509:
7505:
7501:
7462:
7454:
7410:
7405:
7381:
7378:
7377:
7373:
7326:
7318:
7294:
7291:
7290:
7286:
7257:
7254:
7253:
7242:
7238:
7234:
7224:
7216:
7195:
7187:
7173:
7153:
7151:
7148:
7147:
7131:
7128:
7127:
7108:
7106:
7042:
7039:
7038:
7022:
7019:
7018:
6981:
6978:
6977:
6973:
6948:
6945:
6944:
6892:
6889:
6888:
6863:
6860:
6859:
6834:
6831:
6830:
6802:
6794:
6783:
6780:
6779:
6754:
6751:
6750:
6719:
6716:
6715:
6666:
6663:
6662:
6634:
6631:
6630:
6602:
6599:
6598:
6573:
6570:
6569:
6539:
6535:
6520:
6516:
6510:
6499:
6481:
6477:
6471:
6467:
6461:
6450:
6445:
6441:
6433:
6430:
6429:
6388:
6384:
6382:
6379:
6378:
6361:
6357:
6355:
6352:
6351:
6335:
6332:
6331:
6313:
6312:
6284:
6263:
6262:
6219:
6194:
6192:
6189:
6188:
6170:
6167:
6166:
6165:and a constant
6147:
6144:
6143:
6127:
6124:
6123:
6091:
6088:
6087:
6078:
6077:
6042:
6039:
6038:
6016:
6013:
6012:
5980:
5976:
5974:
5971:
5970:
5948:
5945:
5944:
5933:
5895:
5891:
5882:
5878:
5872:
5868:
5858:
5850:
5848:
5842:
5834:
5820:
5818:
5815:
5814:
5787:
5783:
5760:
5752:
5749:
5748:
5723:
5722:
5702:
5700:
5694:
5693:
5679:
5677:
5664:
5656:
5654:
5647:
5646:
5627:
5623:
5617:
5613:
5604:
5599:
5593:
5590:
5589:
5551:
5543:
5540:
5539:
5497:
5491:
5487:
5483:
5479:
5470:
5462:
5443:
5441:
5438:
5437:
5399:
5396:
5395:
5391:Standard Normal
5355:
5335:
5333:
5329:
5328:
5318:
5314:
5310:
5300:
5292:
5279:
5275:
5263:
5261:
5258:
5257:
5236:
5232:
5212:
5209:
5208:
5179:
5157:
5153:
5141:
5136:
5130:
5127:
5126:
5091:
5088:
5087:
5052:
5050:
5027:
5022:
5016:
5011:
5005:
5002:
5001:
4963:
4960:
4959:
4930:
4915:
4911:
4887:
4876:
4870:
4867:
4866:
4816:
4808:
4805:
4804:
4769:
4762:
4758:
4749:
4745:
4741:
4739:
4733:
4722:
4716:
4713:
4712:
4683:
4675:
4672:
4671:
4631:
4627:
4609:
4605:
4598:
4585:
4584:
4583:
4574:
4563:
4557:
4554:
4553:
4515:
4512:
4511:
4451:
4448:
4447:
4406:
4403:
4402:
4374:
4361:
4358:Harmonic series
4353:
4349:
4345:
4341:
4333:
4326:
4320:
4316:
4312:
4304:
4296:
4290:
4286:
4282:
4278:
4271:
4264:
4251:
4236:
4232:
4228:
4224:
4205:
4204:
4186:
4182:
4170: and
4168:
4153:
4149:
4135:
4133:
4128:
4125:
4124:
4106:
4102:
4090: and
4088:
4073:
4069:
4055:
4053:
4044:
4043:
4025:
4021:
4009: and
4007:
3992:
3988:
3974:
3972:
3963:
3962:
3944:
3940:
3928: and
3926:
3911:
3907:
3893:
3891:
3882:
3878:
3857:
3853:
3837:
3836:
3816:
3813:
3812:
3808:
3804:
3800:
3786:
3775:
3764:
3756:
3752:
3740:
3680:
3661:
3642:
3623:
3608:
3604:
3597:
3593:
3587:
3576:
3552:
3549:
3548:
3544:
3540:
3532:
3528:
3520:
3512:
3509:
3465:
3457:
3437:
3436:
3412:
3411:
3405:
3400:
3376:
3373:
3372:
3344:
3343:
3319:
3318:
3312:
3307:
3275:
3267:
3261:
3258:
3257:
3202:
3161:
3158:
3157:
3141:
3138:
3137:
3121:
3118:
3117:
3101:
3098:
3097:
3063:
3060:
3059:
3043:
3040:
3039:
3020:
3017:
3016:
3009:
3005:
3004:Expected value
2992:
2952:
2951:
2947:
2928:
2924:
2904:
2901:
2900:
2892:
2884:
2876:
2872:
2868:
2864:
2860:
2856:
2852:
2845:
2841:
2833:
2822:
2811:
2769:
2765:
2739:
2736:
2735:
2731:
2716:
2711:
2703:
2672:
2668:
2648:
2645:
2644:
2637:
2633:
2629:
2623:random variable
2618:
2611:
2602:
2598:
2594:
2583:
2572:
2562:
2555:
2548:
2525:
2521:
2512:
2508:
2507:
2500:
2496:
2487:
2483:
2482:
2480:
2464:
2445:
2441:
2432:
2428:
2427:
2422:
2416:
2411:
2376:
2371:
2365:
2362:
2361:
2346:
2342:
2335:
2266:
2258:
2234:
2231:
2230:
2226:
2222:
2210:
2202:
2194:
2191:
2158:
2152:
2121:
2106:
2091:
2065:
2041:
2017:
2002:
1998:
1992:
1988:
1982:
1957:
1954:
1953:
1928:
1922:
1914:
1911:
1910:
1867:
1864:
1863:
1842:
1838:
1831:
1825:
1816:
1812:
1810:
1807:
1806:
1783:
1779:
1777:
1774:
1773:
1769:
1738:
1731:
1725:
1721:
1717:
1710:
1704:
1684:
1680:
1673:
1669:
1663:
1652:
1628:
1625:
1624:
1617:
1604:
1601:
1598:
1597:
1595:
1571:
1552:
1530:
1506:
1497:gain from
1495:
1483:
1480:
1479:
1472:
1469:
1466:
1465:
1463:
1447:
1444:
1443:
1408:
1405:
1404:
1401:arithmetic mean
1384:
1381:
1380:
1350:
1331:
1312:
1293:
1274:
1255:
1229:
1226:
1225:
1209:
1206:
1205:
1198:
1195:
1192:
1191:
1189:
1173:
1170:
1169:
1145:
1142:
1141:
1121:
1118:
1117:
1105:
1093:
1084:
1078:
1068:
1060:
1057:
1049:
1041:
1037:
1028:
1022:
1000:
996:
990:
986:
971:
967:
961:
957:
948:
944:
938:
934:
914:
911:
910:
906:
903:
894:
888:
885:
876:
870:
862:
859:
850:
844:
841:
833:
829:
822:
816:
812:
790:
777:
756:
754:
751:
750:
747:
739:
731:
728:
722:
712:
708:
700:
697:blackboard bold
680:
678:
675:
674:
670:
666:
648:
645:W. A. Whitworth
640:
637:
604:
587:
584:
526:
515:
499:
497:
494:
493:
489:
482:
478:
470:
466:
444:random variable
389:
237:Random variable
188:Bernoulli trial
69:
48:
44:
42:
39:
38:
35:
28:
23:
22:
15:
12:
11:
5:
14113:
14103:
14102:
14097:
14080:
14079:
14077:
14076:
14071:
14066:
14060:
14055:
14048:
14045:
14044:
14042:
14041:
14036:
14031:
14026:
14021:
14016:
14011:
14009:central moment
14006:
14000:
13997:
13996:
13989:
13987:
13985:
13984:
13979:
13973:
13967:
13960:
13957:
13956:
13948:
13947:
13940:
13933:
13925:
13918:
13917:
13903:
13879:
13867:
13847:
13833:
13813:
13793:
13773:
13767:
13747:
13733:
13717:
13703:
13697:
13683:
13681:
13678:
13675:
13674:
13662:
13650:
13648:, Section V.8.
13631:
13619:
13607:
13605:, Section 2.8.
13580:
13578:, Section 6-4.
13568:
13566:, Section V.6.
13556:
13526:
13514:
13502:
13500:, p. 103.
13486:
13470:
13450:
13444:, p. 99;
13434:
13428:, p. 97;
13418:
13412:, p. 92;
13398:
13382:
13376:, p. 89;
13366:
13330:
13318:
13306:
13294:
13279:
13277:, p. 273.
13267:
13255:
13253:, Section I.2.
13243:
13227:
13212:
13200:
13188:
13186:, p. 221.
13176:
13162:
13149:
13108:
13090:
13083:. New Series.
13071:
13052:(5): 409β419.
13036:
13029:
13004:
12997:
12979:
12947:
12922:
12921:
12919:
12916:
12915:
12914:
12908:
12903:
12898:
12892:
12877:
12863:
12846:
12840:
12835:
12830:
12823:
12820:
12805:
12801:
12794:
12791:
12785:
12782:
12779:
12774:
12767:
12764:
12757:
12754:
12749:
12745:
12741:
12738:
12735:
12712:
12709:
12682:
12679:
12676:
12672:
12665:
12662:
12655:
12651:
12648:
12645:
12642:
12636:
12633:
12627:
12618:is written as
12607:
12604:
12600:
12572:
12569:
12537:
12532:
12528:
12524:
12521:
12518:
12515:
12512:
12509:
12506:
12503:
12498:
12494:
12490:
12487:
12484:
12481:
12478:
12475:
12472:
12469:
12466:
12450:
12443:
12435:
12428:
12419:center of mass
12392:
12387:
12362:
12355:
12332:
12329:
12323:
12316:
12310:
12307:
12304:
12301:
12298:
12292:
12287:
12284:
12280:
12277:
12274:
12131:
12128:
12111:
12108:
12105:
12102:
12099:
12096:
12093:
12090:
12070:
12067:
12064:
12061:
12058:
12034:
12031:
12025:
12022:
12019:
12016:
12012:
12008:
12005:
12002:
11999:
11993:
11988:
11984:
11981:
11978:
11975:
11972:
11952:
11949:
11946:
11942:
11939:
11936:
11931:
11927:
11923:
11920:
11917:
11914:
11908:
11903:
11896:
11893:
11889:
11884:
11881:
11878:
11875:
11872:
11869:
11866:
11863:
11860:
11836:
11833:
11830:
11827:
11824:
11821:
11818:
11815:
11793:
11790:
11787:
11783:
11779:
11776:
11772:
11769:
11766:
11761:
11757:
11751:
11748:
11745:
11742:
11738:
11731:
11726:
11721:
11717:
11714:
11711:
11708:
11702:
11697:
11690:
11687:
11683:
11678:
11675:
11672:
11669:
11666:
11663:
11660:
11657:
11654:
11643:Borel function
11628:
11623:
11618:
11613:
11610:
11590:
11587:
11584:
11581:
11559:
11556:
11553:
11549:
11546:
11543:
11538:
11534:
11528:
11525:
11522:
11519:
11515:
11508:
11503:
11496:
11493:
11489:
11484:
11481:
11478:
11475:
11470:
11466:
11443:
11439:
11415:
11393:
11389:
11376:
11373:
11372:
11371:
11356:
11351:
11347:
11343:
11323:
11318:
11314:
11308:
11304:
11300:
11297:
11294:
11291:
11288:
11283:
11279:
11275:
11272:
11269:
11264:
11260:
11245:
11244:
11243:
11232:
11229:
11225:
11221:
11218:
11213:
11209:
11204:
11200:
11197:
11192:
11188:
11177:
11166:
11163:
11160:
11157:
11154:
11151:
11148:
11143:
11139:
11135:
11132:
11129:
11124:
11120:
11109:
11097:
11094:
11091:
11088:
11085:
11082:
11079:
11076:
11072:
11068:
11064:
11060:
11057:
11035:
11032:
11029:
11026:
11023:
11020:
11017:
11014:
10994:
10991:
10988:
10985:
10982:
10978:
10972:
10968:
10963:
10939:
10936:
10931:
10927:
10906:
10903:
10900:
10897:
10894:
10889:
10885:
10881:
10867:
10853:
10849:
10843:
10839:
10838:lim inf
10835:
10832:
10809:
10806:
10803:
10800:
10797:
10794:
10791:
10788:
10768:
10765:
10760:
10756:
10735:
10732:
10729:
10709:
10706:
10703:
10698:
10694:
10690:
10687:
10684:
10664:
10661:
10656:
10652:
10628:
10625:
10620:
10616:
10612:
10609:
10606:
10601:
10597:
10596:lim inf
10593:
10590:
10585:
10581:
10575:
10571:
10570:lim inf
10567:
10564:
10561:
10541:
10538:
10535:
10532:
10529:
10526:
10523:
10518:
10514:
10510:
10496:
10485:
10482:
10477:
10473:
10469:
10466:
10463:
10458:
10453:
10450:
10447:
10443:
10439:
10435:
10429:
10425:
10419:
10414:
10411:
10408:
10404:
10399:
10395:
10392:
10366:
10361:
10358:
10355:
10351:
10345:
10341:
10337:
10317:
10314:
10311:
10308:
10305:
10302:
10299:
10296:
10291:
10287:
10283:
10280:
10277:
10272:
10268:
10247:
10244:
10239:
10235:
10214:
10211:
10208:
10186:
10183:
10180:
10176:
10172:
10167:
10163:
10159:
10156:
10136:
10133:
10130:
10127:
10124:
10119:
10115:
10111:
10079:
10075:
10054:
10051:
10048:
10045:
10023:
10019:
10015:
10010:
10007:
10004:
10000:
9996:
9991:
9987:
9964:
9960:
9956:
9951:
9947:
9924:
9921:
9916:
9912:
9908:
9905:
9902:
9897:
9892:
9889:
9886:
9882:
9878:
9874:
9868:
9864:
9858:
9853:
9850:
9847:
9843:
9838:
9834:
9831:
9811:
9791:
9788:
9785:
9782:
9779:
9776:
9771:
9767:
9763:
9741:
9737:
9731:
9727:
9721:
9718:
9715:
9711:
9706:
9702:
9699:
9696:
9693:
9690:
9687:
9684:
9681:
9676:
9672:
9668:
9665:
9662:
9657:
9654:
9651:
9647:
9626:
9623:
9603:
9600:
9594:
9591:
9585:
9582:
9579:
9575:
9570:
9563:
9560:
9554:
9551:
9547:
9543:
9540:
9536:
9532:
9529:
9526:
9523:
9520:
9515:
9511:
9507:
9504:
9501:
9481:
9478:
9473:
9469:
9448:
9445:
9425:
9422:
9419:
9415:
9394:
9390:
9385:
9378:
9375:
9369:
9366:
9362:
9358:
9355:
9351:
9346:
9342:
9339:
9336:
9331:
9327:
9306:
9303:
9300:
9297:
9277:
9274:
9271:
9268:
9265:
9262:
9242:
9222:
9219:
9214:
9210:
9189:
9186:
9183:
9180:
9177:
9174:
9171:
9166:
9162:
9158:
9155:
9152:
9140:
9137:
9133:measure spaces
9129:
9128:
9117:
9112:
9108:
9104:
9098:
9090:
9085:
9080:
9076:
9072:
9069:
9064:
9059:
9054:
9050:
9046:
9040:
9032:
9027:
9022:
9018:
9014:
9011:
9006:
9001:
8996:
8992:
8988:
8982:
8974:
8969:
8964:
8961:
8958:
8954:
8950:
8947:
8942:
8883:
8878:is called the
8848:
8843:
8839:
8835:
8831:
8825:
8820:
8815:
8811:
8807:
8804:
8801:
8796:
8792:
8788:
8784:
8778:
8773:
8768:
8764:
8760:
8757:
8754:
8751:
8747:
8743:
8740:
8736:
8732:
8729:
8690:
8664:
8659:
8655:
8651:
8646:
8640:
8635:
8630:
8626:
8622:
8619:
8615:
8610:
8605:
8601:
8597:
8592:
8586:
8581:
8576:
8572:
8568:
8565:
8561:
8488:
8485:
8482:
8479:
8476:
8473:
8470:
8467:
8464:
8461:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8437:
8364:
8357:
8353:
8348:
8345:
8342:
8339:
8336:
8330:
8327:
8324:
8321:
8317:
8313:
8310:
8307:
8299:
8296:
8292:
8288:
8285:
8282:
8245:
8240:
8236:
8233:
8230:
8227:
8224:
8218:
8215:
8212:
8209:
8206:
8203:
8200:
8197:
8166:
8163:
8162:
8161:
8145:
8125:
8105:
8102:
8099:
8095:
8092:
8089:
8086:
8083:
8080:
8077:
8074:
8068:
8063:
8059:
8056:
8053:
8050:
8047:
8044:
8041:
8038:
8035:
8015:
7995:
7971:
7968:
7965:
7962:
7959:
7939:
7919:
7916:
7913:
7910:
7907:
7888:
7885:
7871:
7859:
7856:
7853:
7850:
7847:
7844:
7841:
7838:
7835:
7832:
7829:
7826:
7823:
7820:
7817:
7814:
7811:
7808:
7784:
7781:
7778:
7775:
7772:
7769:
7766:
7763:
7760:
7757:
7754:
7751:
7748:
7745:
7742:
7739:
7736:
7733:
7709:
7689:
7669:
7666:
7663:
7660:
7657:
7654:
7651:
7648:
7645:
7642:
7639:
7636:
7616:
7613:
7610:
7607:
7604:
7601:
7590:
7574:
7571:
7568:
7565:
7562:
7559:
7556:
7551:
7546:
7543:
7540:
7536:
7532:
7529:
7526:
7523:
7520:
7517:
7489:
7486:
7483:
7479:
7476:
7473:
7470:
7465:
7460:
7457:
7453:
7449:
7446:
7443:
7439:
7436:
7433:
7430:
7427:
7424:
7421:
7418:
7413:
7408:
7404:
7400:
7397:
7394:
7391:
7388:
7385:
7353:
7350:
7347:
7344:
7341:
7338:
7334:
7329:
7324:
7321:
7317:
7313:
7310:
7307:
7304:
7301:
7298:
7270:
7267:
7264:
7261:
7250:
7220:
7213:
7202:
7198:
7194:
7190:
7186:
7183:
7180:
7176:
7172:
7169:
7166:
7163:
7160:
7156:
7135:
7103:
7091:
7088:
7085:
7082:
7079:
7076:
7073:
7070:
7067:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7026:
7006:
7003:
7000:
6997:
6994:
6991:
6988:
6985:
6958:
6955:
6952:
6941:
6929:
6926:
6923:
6920:
6917:
6914:
6911:
6908:
6905:
6902:
6899:
6896:
6873:
6870:
6867:
6856:
6844:
6841:
6838:
6818:
6815:
6812:
6809:
6805:
6801:
6797:
6793:
6790:
6787:
6776:
6764:
6761:
6758:
6738:
6735:
6732:
6729:
6726:
6723:
6703:
6700:
6697:
6694:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6650:
6647:
6644:
6641:
6638:
6618:
6615:
6612:
6609:
6606:
6583:
6580:
6577:
6566:
6550:
6547:
6542:
6538:
6534:
6531:
6528:
6523:
6519:
6513:
6508:
6505:
6502:
6498:
6494:
6490:
6484:
6480:
6474:
6470:
6464:
6459:
6456:
6453:
6449:
6444:
6440:
6437:
6417:
6414:
6411:
6408:
6405:
6402:
6399:
6396:
6391:
6387:
6377:and constants
6364:
6360:
6339:
6311:
6308:
6305:
6302:
6299:
6296:
6293:
6290:
6287:
6285:
6283:
6280:
6277:
6274:
6271:
6268:
6265:
6264:
6261:
6258:
6255:
6252:
6249:
6246:
6243:
6240:
6237:
6234:
6231:
6228:
6225:
6222:
6220:
6218:
6215:
6212:
6209:
6206:
6203:
6200:
6197:
6196:
6177:
6174:
6154:
6151:
6131:
6107:
6104:
6101:
6098:
6095:
6075:
6064:
6061:
6058:
6055:
6052:
6049:
6046:
6026:
6023:
6020:
5997:
5993:
5989:
5986:
5983:
5979:
5958:
5955:
5952:
5932:
5929:
5926:
5925:
5910:
5907:
5898:
5894:
5890:
5885:
5881:
5875:
5871:
5867:
5864:
5861:
5856:
5853:
5845:
5840:
5837:
5833:
5827:
5824:
5812:
5801:
5798:
5795:
5790:
5786:
5782:
5778:
5775:
5772:
5769:
5766:
5763:
5759:
5756:
5746:
5740:
5739:
5726:
5721:
5718:
5715:
5712:
5709:
5701:
5699:
5696:
5695:
5692:
5689:
5686:
5678:
5673:
5670:
5667:
5662:
5659:
5653:
5652:
5650:
5645:
5642:
5639:
5633:
5630:
5626:
5620:
5616:
5612:
5607:
5602:
5598:
5587:
5576:
5573:
5570:
5567:
5564:
5560:
5557:
5554:
5550:
5547:
5537:
5531:
5530:
5519:
5516:
5513:
5510:
5504:
5500:
5494:
5490:
5486:
5482:
5478:
5473:
5468:
5465:
5461:
5454:
5451:
5447:
5435:
5424:
5421:
5418:
5415:
5412:
5409:
5406:
5403:
5393:
5387:
5386:
5375:
5372:
5369:
5366:
5358:
5353:
5348:
5344:
5341:
5338:
5332:
5325:
5322:
5317:
5313:
5308:
5303:
5298:
5295:
5291:
5282:
5278:
5274:
5271:
5267:
5255:
5244:
5239:
5235:
5231:
5228:
5225:
5222:
5219:
5216:
5206:
5200:
5199:
5186:
5183:
5178:
5175:
5172:
5166:
5163:
5160:
5156:
5152:
5149:
5144:
5139:
5135:
5124:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5085:
5079:
5078:
5065:
5061:
5058:
5055:
5049:
5046:
5043:
5036:
5033:
5030:
5026:
5019:
5014:
5010:
4999:
4988:
4985:
4982:
4979:
4976:
4973:
4970:
4967:
4957:
4951:
4950:
4937:
4934:
4929:
4924:
4921:
4918:
4914:
4910:
4907:
4904:
4901:
4898:
4895:
4890:
4885:
4882:
4879:
4875:
4864:
4853:
4850:
4847:
4843:
4840:
4837:
4834:
4831:
4828:
4825:
4822:
4819:
4815:
4812:
4802:
4796:
4795:
4784:
4781:
4775:
4772:
4765:
4761:
4755:
4752:
4748:
4744:
4736:
4731:
4728:
4725:
4721:
4710:
4699:
4696:
4693:
4689:
4686:
4682:
4679:
4669:
4663:
4662:
4651:
4648:
4645:
4640:
4637:
4634:
4630:
4626:
4623:
4620:
4617:
4612:
4608:
4601:
4596:
4593:
4588:
4582:
4577:
4572:
4569:
4566:
4562:
4551:
4540:
4537:
4534:
4531:
4528:
4525:
4522:
4519:
4509:
4503:
4502:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4445:
4434:
4431:
4428:
4425:
4422:
4419:
4413:
4410:
4400:
4394:
4393:
4390:
4387:
4373:
4370:
4369:
4368:
4365:
4275:
4208:
4203:
4200:
4197:
4194:
4189:
4185:
4181:
4178:
4175:
4167:
4164:
4161:
4156:
4152:
4148:
4145:
4142:
4134:
4127:
4126:
4123:
4120:
4117:
4114:
4109:
4105:
4101:
4098:
4095:
4087:
4084:
4081:
4076:
4072:
4068:
4065:
4062:
4054:
4052:
4049:
4046:
4045:
4042:
4039:
4036:
4033:
4028:
4024:
4020:
4017:
4014:
4006:
4003:
4000:
3995:
3991:
3987:
3984:
3981:
3973:
3971:
3968:
3965:
3964:
3961:
3958:
3955:
3952:
3947:
3943:
3939:
3936:
3933:
3925:
3922:
3919:
3914:
3910:
3906:
3903:
3900:
3892:
3890:
3885:
3881:
3877:
3874:
3871:
3868:
3865:
3860:
3856:
3852:
3849:
3846:
3843:
3842:
3840:
3835:
3832:
3829:
3826:
3823:
3820:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3687:
3684:
3679:
3676:
3673:
3668:
3665:
3660:
3657:
3654:
3649:
3646:
3641:
3638:
3635:
3630:
3627:
3622:
3619:
3616:
3611:
3607:
3600:
3596:
3590:
3585:
3582:
3579:
3575:
3571:
3568:
3565:
3562:
3559:
3556:
3536:
3524:
3508:
3505:
3492:
3489:
3486:
3482:
3479:
3476:
3473:
3468:
3463:
3460:
3456:
3452:
3449:
3446:
3440:
3435:
3432:
3429:
3426:
3423:
3420:
3415:
3408:
3403:
3399:
3395:
3392:
3389:
3386:
3383:
3380:
3356:
3353:
3347:
3342:
3339:
3336:
3333:
3330:
3327:
3322:
3315:
3310:
3306:
3302:
3299:
3296:
3292:
3289:
3286:
3283:
3278:
3273:
3270:
3266:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3219:
3216:
3213:
3210:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3174:
3171:
3168:
3165:
3145:
3125:
3105:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3047:
3024:
2980:
2977:
2973:
2970:
2967:
2964:
2961:
2955:
2950:
2946:
2943:
2940:
2936:
2931:
2927:
2923:
2920:
2917:
2914:
2911:
2908:
2887:is called the
2881:
2880:
2859:such that: if
2849:
2830:
2815:
2796:
2793:
2790:
2786:
2783:
2780:
2777:
2772:
2768:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2719:is said to be
2708:approximations
2690:
2687:
2684:
2680:
2675:
2671:
2667:
2664:
2661:
2658:
2655:
2652:
2615:measure theory
2610:
2607:
2536:
2528:
2524:
2520:
2515:
2511:
2503:
2499:
2495:
2490:
2486:
2479:
2476:
2471:
2468:
2463:
2460:
2457:
2448:
2444:
2440:
2435:
2431:
2426:
2419:
2414:
2410:
2406:
2403:
2400:
2396:
2393:
2390:
2387:
2384:
2379:
2374:
2370:
2309:measure theory
2296:
2293:
2290:
2286:
2283:
2280:
2277:
2274:
2269:
2264:
2261:
2257:
2253:
2250:
2247:
2244:
2241:
2238:
2190:
2187:
2186:
2185:
2174:
2167:
2164:
2161:
2157:
2150:
2146:
2142:
2138:
2135:
2129:
2126:
2120:
2114:
2111:
2105:
2099:
2096:
2089:
2085:
2082:
2079:
2073:
2070:
2064:
2061:
2058:
2055:
2049:
2046:
2040:
2037:
2034:
2031:
2025:
2022:
2016:
2013:
2010:
2005:
2001:
1995:
1991:
1985:
1981:
1977:
1973:
1970:
1967:
1964:
1961:
1937:
1934:
1931:
1927:
1921:
1918:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1845:
1841:
1837:
1834:
1830:
1824:
1819:
1815:
1794:
1791:
1786:
1782:
1768:
1765:
1736:
1729:
1715:
1708:
1692:
1687:
1683:
1676:
1672:
1666:
1661:
1658:
1655:
1651:
1647:
1644:
1641:
1638:
1635:
1632:
1616:
1613:
1612:
1611:
1583:
1578:
1575:
1570:
1567:
1564:
1559:
1556:
1551:
1548:
1545:
1542:
1537:
1534:
1529:
1526:
1523:
1520:
1517:
1514:
1505:
1502:
1493:
1490:
1487:
1451:
1435:
1412:
1388:
1365:
1362:
1357:
1354:
1349:
1346:
1343:
1338:
1335:
1330:
1327:
1324:
1319:
1316:
1311:
1308:
1305:
1300:
1297:
1292:
1289:
1286:
1281:
1278:
1273:
1270:
1267:
1262:
1259:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1213:
1177:
1149:
1125:
1104:
1101:
1089:
1082:
1064:
1053:
1033:
1026:
1008:
1003:
999:
993:
989:
985:
982:
979:
974:
970:
964:
960:
956:
951:
947:
941:
937:
933:
930:
927:
924:
921:
918:
909:is defined as
899:
892:
881:
874:
858:
855:
846:
837:
818:
799:measure theory
789:
786:
763:
760:
745:
724:
683:
653:Erwartungswert
636:
633:
603:
600:
585:Edwards (2002)
582:
562:Huygens (1657)
525:
522:
502:
456:measure theory
401:expected value
391:
390:
388:
387:
380:
373:
365:
362:
361:
360:
359:
354:
346:
345:
344:
343:
338:
336:Bayes' theorem
333:
328:
323:
318:
310:
309:
308:
307:
302:
297:
292:
284:
283:
282:
281:
280:
279:
274:
269:
267:Observed value
264:
259:
254:
252:Expected value
249:
244:
234:
229:
228:
227:
222:
217:
212:
207:
202:
192:
191:
190:
180:
179:
178:
173:
168:
163:
158:
148:
143:
135:
134:
133:
132:
127:
122:
121:
120:
110:
109:
108:
95:
94:
86:
85:
79:
78:
64:function, see
51:
47:
26:
9:
6:
4:
3:
2:
14112:
14101:
14098:
14096:
14093:
14092:
14090:
14075:
14072:
14070:
14067:
14064:
14061:
14059:
14056:
14053:
14050:
14049:
14046:
14040:
14037:
14035:
14032:
14030:
14027:
14025:
14022:
14020:
14017:
14015:
14012:
14010:
14007:
14005:
14002:
14001:
13998:
13993:
13983:
13980:
13977:
13974:
13971:
13968:
13965:
13962:
13961:
13958:
13954:
13946:
13941:
13939:
13934:
13932:
13927:
13926:
13923:
13914:
13910:
13906:
13900:
13896:
13892:
13888:
13884:
13880:
13876:
13870:
13868:0-07-366011-6
13864:
13860:
13856:
13852:
13848:
13844:
13840:
13836:
13834:0-471-58495-9
13830:
13826:
13822:
13818:
13814:
13810:
13806:
13802:
13798:
13794:
13790:
13786:
13782:
13778:
13774:
13770:
13768:0-534-11958-1
13764:
13760:
13756:
13752:
13748:
13744:
13740:
13736:
13734:0-471-00710-2
13730:
13726:
13722:
13718:
13711:
13710:
13704:
13700:
13698:0-8018-6946-3
13694:
13690:
13685:
13684:
13672:, Section 19.
13671:
13666:
13659:
13654:
13647:
13642:
13640:
13638:
13636:
13628:
13623:
13616:
13611:
13604:
13600:
13596:
13592:
13587:
13585:
13577:
13572:
13565:
13560:
13546:
13542:
13535:
13533:
13531:
13523:
13518:
13512:, Chapter 20.
13511:
13506:
13499:
13495:
13490:
13483:
13479:
13474:
13467:
13463:
13459:
13454:
13447:
13443:
13438:
13431:
13427:
13422:
13415:
13411:
13407:
13402:
13395:
13391:
13386:
13379:
13375:
13370:
13360:
13355:
13347:
13343:
13342:
13334:
13327:
13322:
13315:
13310:
13303:
13298:
13292:, Section 15.
13291:
13286:
13284:
13276:
13271:
13264:
13259:
13252:
13247:
13240:
13236:
13231:
13224:
13219:
13217:
13209:
13204:
13198:, p. 76.
13197:
13192:
13185:
13180:
13172:
13166:
13159:
13153:
13145:
13141:
13135:
13127:
13123:
13119:
13112:
13101:
13094:
13086:
13082:
13075:
13067:
13063:
13059:
13055:
13051:
13047:
13040:
13032:
13030:9780471725169
13026:
13022:
13018:
13014:
13008:
13000:
12998:9781441923226
12994:
12990:
12983:
12969:on 2022-01-19
12965:
12958:
12951:
12937:
12933:
12927:
12923:
12912:
12909:
12907:
12904:
12902:
12899:
12896:
12893:
12891:
12890:drawing above
12875:
12867:
12864:
12862:
12858:
12854:
12850:
12847:
12844:
12841:
12839:
12836:
12834:
12831:
12829:
12826:
12825:
12819:
12803:
12789:
12780:
12772:
12762:
12752:
12747:
12739:
12707:
12696:
12680:
12674:
12660:
12649:
12643:
12631:
12602:
12589:
12588:quantum state
12567:
12557:
12553:
12548:
12535:
12530:
12519:
12513:
12504:
12496:
12492:
12485:
12479:
12473:
12467:
12464:
12455:
12453:
12446:
12439:
12431:
12424:
12420:
12416:
12407:
12403:
12390:
12330:
12305:
12299:
12285:
12282:
12275:
12264:
12260:
12256:
12251:
12249:
12245:
12241:
12237:
12232:
12230:
12224:
12219:
12215:
12211:
12207:
12203:
12198:
12196:
12192:
12188:
12183:
12181:
12177:
12171:
12166:
12164:
12160:
12156:
12152:
12148:
12144:
12140:
12135:
12127:
12125:
12103:
12097:
12091:
12068:
12062:
12056:
12048:
12032:
12029:
12023:
12020:
12017:
12014:
12010:
12003:
11997:
11986:
11982:
11976:
11970:
11950:
11947:
11944:
11937:
11929:
11925:
11918:
11912:
11901:
11894:
11891:
11887:
11882:
11873:
11867:
11861:
11850:
11828:
11822:
11816:
11804:
11791:
11788:
11785:
11781:
11777:
11774:
11767:
11759:
11755:
11749:
11746:
11743:
11740:
11736:
11724:
11719:
11712:
11706:
11695:
11688:
11685:
11681:
11676:
11667:
11661:
11655:
11644:
11611:
11608:
11585:
11579:
11570:
11557:
11554:
11551:
11544:
11536:
11532:
11526:
11523:
11520:
11517:
11513:
11501:
11494:
11491:
11487:
11482:
11476:
11468:
11464:
11441:
11437:
11429:
11413:
11391:
11387:
11370:
11349:
11345:
11316:
11312:
11306:
11295:
11289:
11281:
11277:
11270:
11262:
11249:
11246:
11230:
11227:
11219:
11216:
11211:
11207:
11198:
11190:
11178:
11161:
11155:
11149:
11141:
11137:
11130:
11122:
11110:
11092:
11086:
11080:
11074:
11066:
11058:
11048:
11047:
11033:
11027:
11021:
11015:
10989:
10986:
10983:
10980:
10970:
10966:
10952:
10937:
10929:
10925:
10901:
10898:
10895:
10892:
10887:
10883:
10871:
10868:
10851:
10847:
10841:
10833:
10830:
10822:
10807:
10804:
10801:
10795:
10789:
10766:
10758:
10754:
10733:
10730:
10727:
10707:
10704:
10696:
10692:
10685:
10662:
10659:
10654:
10650:
10641:
10626:
10618:
10614:
10607:
10599:
10591:
10583:
10579:
10573:
10562:
10536:
10533:
10530:
10527:
10524:
10521:
10516:
10512:
10500:
10499:Fatou's lemma
10497:
10483:
10475:
10471:
10464:
10451:
10448:
10445:
10441:
10437:
10433:
10427:
10423:
10412:
10409:
10406:
10402:
10397:
10393:
10382:
10359:
10356:
10353:
10343:
10339:
10315:
10309:
10303:
10297:
10289:
10285:
10278:
10270:
10245:
10237:
10233:
10212:
10209:
10206:
10184:
10181:
10178:
10174:
10170:
10165:
10161:
10157:
10154:
10131:
10128:
10125:
10122:
10117:
10113:
10101:
10098:
10097:
10096:
10093:
10077:
10073:
10052:
10049:
10046:
10043:
10021:
10017:
10013:
10008:
10005:
10002:
9998:
9994:
9989:
9985:
9962:
9958:
9954:
9949:
9945:
9935:
9922:
9914:
9910:
9903:
9890:
9887:
9884:
9880:
9876:
9872:
9866:
9862:
9851:
9848:
9845:
9841:
9836:
9832:
9809:
9789:
9783:
9780:
9777:
9774:
9769:
9765:
9752:
9739:
9735:
9729:
9725:
9713:
9704:
9700:
9694:
9691:
9688:
9685:
9682:
9674:
9670:
9663:
9649:
9624:
9621:
9601:
9598:
9592:
9589:
9583:
9580:
9577:
9573:
9568:
9561:
9558:
9552:
9549:
9545:
9541:
9538:
9534:
9527:
9524:
9521:
9513:
9509:
9502:
9479:
9471:
9467:
9446:
9443:
9420:
9392:
9388:
9383:
9376:
9373:
9367:
9364:
9360:
9356:
9353:
9349:
9340:
9337:
9334:
9329:
9325:
9304:
9301:
9298:
9295:
9275:
9269:
9266:
9263:
9240:
9220:
9212:
9208:
9184:
9178:
9164:
9160:
9153:
9136:
9134:
9115:
9110:
9106:
9102:
9088:
9078:
9070:
9057:
9052:
9048:
9044:
9030:
9020:
9012:
8999:
8994:
8990:
8986:
8972:
8962:
8959:
8956:
8948:
8928:
8924:
8916:
8908:
8892:
8887:
8884:
8881:
8875:
8871:
8846:
8841:
8837:
8833:
8823:
8813:
8805:
8794:
8790:
8786:
8776:
8766:
8758:
8749:
8741:
8738:
8730:
8717:
8713:
8706:
8699:
8694:
8691:
8688:
8684:
8662:
8657:
8653:
8649:
8644:
8638:
8628:
8620:
8613:
8608:
8603:
8599:
8595:
8590:
8584:
8574:
8566:
8559:
8548:
8544:
8536:
8531:
8527:
8521:
8511:
8507:
8502:
8501:negative part
8486:
8477:
8471:
8465:
8459:
8450:
8444:
8435:
8423:
8418:
8414:
8410:
8405:
8402:
8401:
8400:
8398:
8393:
8391:
8386:
8382:
8362:
8355:
8351:
8343:
8337:
8334:
8328:
8322:
8319:
8308:
8297:
8294:
8283:
8273:
8267:
8256:
8243:
8238:
8231:
8225:
8216:
8210:
8207:
8204:
8198:
8179:
8175:
8171:
8159:
8158:joint density
8143:
8123:
8103:
8100:
8097:
8090:
8084:
8078:
8072:
8061:
8057:
8048:
8042:
8036:
8013:
7993:
7985:
7984:inner product
7969:
7963:
7957:
7937:
7917:
7911:
7905:
7886:
7883:
7875:
7872:
7857:
7851:
7845:
7836:
7830:
7824:
7818:
7815:
7809:
7798:
7782:
7776:
7770:
7761:
7755:
7749:
7743:
7740:
7734:
7723:
7707:
7687:
7667:
7661:
7655:
7649:
7643:
7637:
7611:
7608:
7602:
7591:
7572:
7566:
7563:
7560:
7544:
7541:
7538:
7534:
7530:
7524:
7518:
7487:
7484:
7481:
7474:
7468:
7463:
7455:
7451:
7447:
7444:
7441:
7431:
7425:
7422:
7419:
7406:
7402:
7398:
7392:
7386:
7371:
7367:
7351:
7345:
7339:
7336:
7332:
7319:
7315:
7311:
7305:
7299:
7284:
7265:
7259:
7251:
7248:
7233:
7229:
7223:
7219:
7214:
7200:
7192:
7184:
7178:
7167:
7161:
7133:
7125:
7120:
7116:
7111:
7104:
7089:
7083:
7077:
7071:
7062:
7056:
7047:
7024:
7004:
7001:
6998:
6992:
6986:
6971:
6956:
6953:
6950:
6942:
6927:
6921:
6915:
6909:
6903:
6897:
6886:
6871:
6868:
6865:
6857:
6842:
6839:
6836:
6816:
6813:
6810:
6799:
6788:
6777:
6762:
6759:
6756:
6736:
6733:
6730:
6727:
6724:
6721:
6701:
6695:
6689:
6683:
6677:
6671:
6645:
6639:
6613:
6607:
6596:
6581:
6578:
6575:
6567:
6564:
6548:
6540:
6536:
6529:
6521:
6517:
6511:
6506:
6503:
6500:
6496:
6492:
6488:
6482:
6478:
6472:
6468:
6462:
6457:
6454:
6451:
6447:
6442:
6438:
6415:
6409:
6406:
6403:
6400:
6397:
6389:
6385:
6362:
6358:
6337:
6329:
6309:
6303:
6297:
6291:
6288:
6286:
6278:
6275:
6269:
6259:
6253:
6247:
6241:
6235:
6229:
6223:
6221:
6213:
6210:
6207:
6201:
6175:
6172:
6152:
6149:
6129:
6121:
6102:
6096:
6085:
6076:
6062:
6059:
6053:
6047:
6037:(a.s.), then
6024:
6021:
6018:
6010:
6009:
6008:
5995:
5991:
5987:
5984:
5981:
5977:
5956:
5953:
5950:
5942:
5941:almost surely
5938:
5924:
5908:
5905:
5896:
5892:
5888:
5883:
5873:
5869:
5865:
5862:
5854:
5851:
5835:
5831:
5825:
5822:
5813:
5796:
5793:
5788:
5784:
5757:
5754:
5747:
5745:
5742:
5741:
5719:
5716:
5713:
5710:
5707:
5690:
5687:
5684:
5671:
5668:
5665:
5660:
5657:
5648:
5643:
5640:
5637:
5631:
5628:
5624:
5618:
5614:
5610:
5600:
5596:
5588:
5571:
5568:
5565:
5548:
5545:
5538:
5536:
5533:
5532:
5517:
5514:
5511:
5508:
5502:
5498:
5492:
5488:
5484:
5480:
5476:
5463:
5459:
5452:
5449:
5445:
5436:
5419:
5416:
5413:
5407:
5404:
5401:
5394:
5392:
5389:
5388:
5373:
5370:
5367:
5364:
5356:
5351:
5346:
5342:
5339:
5336:
5330:
5323:
5320:
5315:
5311:
5306:
5293:
5289:
5280:
5276:
5272:
5269:
5265:
5256:
5237:
5233:
5229:
5226:
5220:
5217:
5214:
5207:
5205:
5202:
5201:
5184:
5181:
5176:
5173:
5170:
5164:
5161:
5158:
5154:
5150:
5147:
5137:
5133:
5125:
5108:
5102:
5099:
5096:
5093:
5086:
5084:
5081:
5080:
5063:
5059:
5056:
5053:
5047:
5044:
5041:
5034:
5031:
5028:
5024:
5017:
5012:
5008:
5000:
4983:
4980:
4977:
4971:
4968:
4965:
4958:
4956:
4953:
4952:
4935:
4932:
4927:
4922:
4919:
4916:
4908:
4905:
4902:
4896:
4893:
4883:
4880:
4877:
4873:
4865:
4848:
4813:
4810:
4803:
4801:
4798:
4797:
4782:
4779:
4773:
4770:
4763:
4759:
4753:
4750:
4746:
4742:
4729:
4726:
4723:
4719:
4711:
4694:
4680:
4677:
4670:
4668:
4665:
4664:
4649:
4646:
4643:
4638:
4635:
4632:
4624:
4621:
4618:
4610:
4606:
4594:
4591:
4580:
4575:
4570:
4567:
4564:
4560:
4552:
4535:
4532:
4529:
4523:
4520:
4517:
4510:
4508:
4505:
4504:
4489:
4486:
4483:
4480:
4477:
4474:
4468:
4465:
4462:
4456:
4453:
4446:
4429:
4426:
4423:
4417:
4411:
4408:
4401:
4399:
4396:
4395:
4391:
4388:
4386:Distribution
4385:
4384:
4381:
4379:
4366:
4364:is undefined.
4359:
4337:
4330:
4323:
4308:
4300:
4293:
4281:takes values
4276:
4267:
4262:
4261:
4260:
4255:
4248:
4244:
4240:
4221:
4201:
4195:
4187:
4183:
4176:
4162:
4154:
4150:
4143:
4121:
4115:
4107:
4103:
4096:
4082:
4074:
4070:
4063:
4047:
4040:
4034:
4026:
4022:
4015:
4001:
3993:
3989:
3982:
3966:
3959:
3953:
3945:
3941:
3934:
3920:
3912:
3908:
3901:
3883:
3879:
3872:
3866:
3858:
3854:
3847:
3838:
3833:
3827:
3821:
3797:
3793:
3789:
3782:
3778:
3771:
3767:
3762:
3750:
3744:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3685:
3682:
3677:
3674:
3671:
3666:
3663:
3658:
3655:
3652:
3647:
3644:
3639:
3636:
3633:
3628:
3625:
3620:
3617:
3614:
3609:
3605:
3598:
3594:
3583:
3580:
3577:
3573:
3569:
3563:
3557:
3539:
3535:
3527:
3523:
3518:
3504:
3490:
3487:
3484:
3477:
3471:
3466:
3458:
3454:
3450:
3447:
3444:
3430:
3424:
3421:
3418:
3401:
3397:
3393:
3387:
3381:
3370:
3354:
3351:
3337:
3331:
3328:
3325:
3308:
3304:
3300:
3297:
3294:
3287:
3281:
3276:
3268:
3264:
3243:
3240:
3237:
3234:
3228:
3222:
3217:
3214:
3211:
3208:
3195:
3189:
3186:
3183:
3180:
3177:
3172:
3169:
3166:
3163:
3143:
3123:
3103:
3083:
3080:
3074:
3068:
3045:
3038:
3022:
3002:
2998:
2996:
2978:
2975:
2968:
2962:
2959:
2948:
2944:
2938:
2934:
2925:
2921:
2915:
2909:
2898:
2890:
2875:is less than
2850:
2839:
2831:
2828:
2820:
2816:
2810:
2794:
2791:
2788:
2781:
2775:
2770:
2766:
2762:
2756:
2753:
2750:
2744:
2730:
2726:
2725:
2724:
2722:
2709:
2688:
2682:
2678:
2669:
2665:
2659:
2653:
2643:
2636:, denoted by
2628:
2625:defined on a
2624:
2616:
2606:
2592:
2587:
2580:
2576:
2569:
2565:
2558:
2551:
2534:
2526:
2522:
2518:
2513:
2509:
2501:
2497:
2493:
2488:
2484:
2477:
2474:
2469:
2466:
2461:
2458:
2455:
2446:
2442:
2438:
2433:
2429:
2424:
2417:
2412:
2408:
2404:
2401:
2398:
2391:
2385:
2382:
2377:
2372:
2368:
2357:
2353:
2349:
2341:
2332:
2330:
2326:
2322:
2318:
2314:
2310:
2294:
2291:
2288:
2281:
2275:
2272:
2259:
2255:
2251:
2245:
2239:
2220:
2216:
2215:open interval
2208:
2200:
2172:
2165:
2162:
2159:
2155:
2148:
2144:
2140:
2136:
2133:
2127:
2124:
2118:
2112:
2109:
2103:
2097:
2094:
2087:
2083:
2080:
2071:
2068:
2059:
2056:
2047:
2044:
2035:
2032:
2023:
2020:
2011:
2008:
2003:
1999:
1993:
1989:
1983:
1979:
1975:
1968:
1962:
1935:
1932:
1929:
1925:
1919:
1916:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1843:
1839:
1835:
1832:
1828:
1822:
1817:
1813:
1792:
1789:
1784:
1780:
1771:
1770:
1764:
1763:
1759:
1754:
1751:
1747:
1742:
1735:
1728:
1714:
1707:
1690:
1685:
1681:
1674:
1670:
1659:
1656:
1653:
1649:
1645:
1639:
1633:
1622:
1581:
1576:
1573:
1565:
1562:
1557:
1554:
1549:
1546:
1540:
1535:
1532:
1527:
1524:
1518:
1515:
1503:
1488:
1449:
1440:
1436:
1433:
1429:
1426:
1425:almost surely
1410:
1402:
1386:
1379:
1363:
1360:
1355:
1352:
1347:
1344:
1341:
1336:
1333:
1328:
1325:
1322:
1317:
1314:
1309:
1306:
1303:
1298:
1295:
1290:
1287:
1284:
1279:
1276:
1271:
1268:
1265:
1260:
1257:
1252:
1249:
1246:
1240:
1234:
1211:
1175:
1167:
1163:
1147:
1139:
1123:
1115:
1114:
1109:
1100:
1098:
1092:
1088:
1081:
1076:
1071:
1067:
1063:
1056:
1052:
1047:
1036:
1032:
1025:
1019:
1006:
1001:
997:
991:
987:
983:
980:
977:
972:
968:
962:
958:
954:
949:
945:
939:
935:
931:
925:
919:
902:
898:
891:
884:
880:
873:
868:
854:
849:
840:
836:
828:
827:random matrix
821:
811:
810:random vector
806:
804:
800:
796:
785:
781:
758:
743:
735:
727:
719:
716:
704:
698:
673:(italic), or
663:
662:
658:
654:
646:
631:
630:
624:
622:
618:
612:
607:
599:
597:
593:
581:
576:
573:
571:
567:
563:
559:
554:
552:
547:
544:
540:
539:Blaise Pascal
535:
534:in a fair way
531:
521:
518:
486:
474:
463:
461:
457:
453:
448:
445:
441:
437:
433:
432:
426:
422:
418:
414:
410:
406:
403:(also called
402:
398:
386:
381:
379:
374:
372:
367:
366:
364:
363:
358:
355:
353:
350:
349:
348:
347:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
317:
314:
313:
312:
311:
306:
303:
301:
298:
296:
293:
291:
288:
287:
286:
285:
278:
275:
273:
270:
268:
265:
263:
260:
258:
255:
253:
250:
248:
245:
243:
240:
239:
238:
235:
233:
230:
226:
223:
221:
218:
216:
213:
211:
208:
206:
203:
201:
198:
197:
196:
193:
189:
186:
185:
184:
181:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
153:
152:
149:
147:
144:
142:
139:
138:
137:
136:
131:
128:
126:
125:Indeterminism
123:
119:
116:
115:
114:
111:
107:
104:
103:
102:
99:
98:
97:
96:
92:
88:
87:
84:
81:
80:
77:
73:
72:
67:
49:
45:
33:
19:
14013:
13886:
13858:
13824:
13821:Kotz, Samuel
13800:
13780:
13758:
13724:
13708:
13688:
13680:Bibliography
13665:
13653:
13622:
13610:
13571:
13559:
13548:. Retrieved
13544:
13517:
13505:
13489:
13473:
13453:
13437:
13421:
13401:
13385:
13369:
13345:
13340:
13333:
13321:
13309:
13297:
13270:
13265:, p. 5.
13258:
13246:
13230:
13203:
13191:
13179:
13165:
13157:
13152:
13117:
13111:
13093:
13084:
13080:
13074:
13049:
13045:
13039:
13012:
13007:
12988:
12982:
12971:. Retrieved
12964:the original
12950:
12939:. Retrieved
12935:
12926:
12860:
12856:
12852:
12549:
12456:
12448:
12441:
12433:
12426:
12422:
12412:
12252:
12233:
12222:
12217:
12209:
12201:
12199:
12184:
12173:
12145:for unknown
12136:
12133:
11805:
11571:
11378:
11368:
11005:(a.s.), and
10820:
10779:(a.s), then
10639:
10094:
9936:
9753:
9142:
9130:
8926:
8922:
8914:
8906:
8890:
8873:
8869:
8715:
8711:
8704:
8697:
8546:
8542:
8534:
8529:
8525:
8519:
8509:
8505:
8416:
8412:
8408:
8394:
8265:
8257:
8177:
8168:
8165:Inequalities
7221:
7217:
7118:
7114:
7109:
6661:exist, then
6083:
5934:
4375:
4335:
4328:
4325:takes value
4321:
4306:
4298:
4295:takes value
4291:
4265:
4253:
4246:
4242:
4238:
4222:
3795:
3791:
3787:
3780:
3776:
3769:
3765:
3748:
3745:
3537:
3533:
3525:
3521:
3510:
3014:
2997:
2888:
2882:
2720:
2707:
2612:
2588:
2578:
2574:
2567:
2563:
2556:
2549:
2355:
2351:
2347:
2343:Cauchy(0, Ο)
2333:
2328:
2327:. Sometimes
2316:
2197:which has a
2192:
1761:
1755:
1743:
1733:
1726:
1712:
1705:
1618:
1090:
1086:
1079:
1075:equiprobable
1072:
1065:
1061:
1054:
1050:
1034:
1030:
1023:
1020:
900:
896:
889:
882:
878:
871:
866:
860:
847:
838:
834:
819:
807:
791:
779:
741:
733:
725:
720:
714:
702:
664:
660:
656:
652:
638:
628:
626:
620:
614:
609:
605:
589:
578:
574:
565:
555:
548:
533:
527:
516:
484:
472:
464:
449:
428:
424:
420:
416:
412:
408:
404:
400:
394:
357:Tree diagram
352:Venn diagram
316:Independence
262:Markov chain
251:
146:Sample space
13646:Feller 1971
13627:Feller 1968
13615:Feller 1968
13595:Feller 1971
13591:Feller 1968
13564:Feller 1971
13522:Feller 1971
13263:Feller 1971
13251:Feller 1971
13223:Feller 1968
13184:Feller 1968
12695:uncertainty
12195:frequencies
12155:sample mean
8178:nonnegative
7930:given that
7722:independent
7226:denote the
6597:, and both
6563:linear form
5083:Exponential
4274:as desired.
3749:nonnegative
3008:and median
669:(upright),
651:stands for
452:integration
405:expectation
272:Random walk
113:Determinism
101:Probability
14089:Categories
14004:raw moment
13951:Theory of
13550:2020-09-11
12973:2021-07-20
12941:2020-09-11
12918:References
12168:See also:
12163:true value
12147:parameters
12139:statistics
10640:Corollary.
8532:) = |
8270:to obtain
7508:, one has
5931:Properties
4392:Mean E(X)
2345:, so that
1077:(that is,
788:Definition
730:, whereas
409:expectancy
183:Experiment
130:Randomness
76:statistics
14074:combinant
13603:Ross 2019
13482:Ross 2019
13466:Ross 2019
13446:Ross 2019
13430:Ross 2019
13414:Ross 2019
13394:Ross 2019
13378:Ross 2019
13239:Ross 2019
13208:Ross 2019
13134:cite book
13087:(1): 549.
12843:Expectile
12800:⟩
12793:^
12784:⟨
12781:−
12778:⟩
12766:^
12756:⟨
12737:Δ
12711:^
12678:⟩
12675:ψ
12664:^
12650:ψ
12647:⟨
12641:⟩
12635:^
12626:⟨
12606:⟩
12603:ψ
12571:^
12514:
12505:−
12486:
12468:
12306:
12286:∈
12276:
12240:residuals
12143:estimates
12092:
12015:−
11987:∫
11926:φ
11902:∫
11895:π
11862:
11817:
11756:φ
11741:−
11725:∫
11696:∫
11689:π
11656:
11622:→
11533:φ
11518:−
11502:∫
11495:π
11438:φ
11296:
11271:
11217:−
11199:
11156:
11131:
11096:∞
11081:
11075:≤
11059:
11031:∞
11016:
10993:∞
10987:≤
10981:≤
10951:pointwise
10935:→
10899:≥
10802:≤
10790:
10764:→
10731:≥
10705:≤
10686:
10660:≥
10608:
10592:≤
10563:
10534:≥
10522:≥
10465:
10457:∞
10442:∑
10418:∞
10403:∑
10394:
10365:∞
10304:
10279:
10243:→
10210:≥
10171:≤
10158:≤
10129:≥
10047:≥
10014:−
9904:
9896:∞
9881:∑
9877:≠
9857:∞
9842:∑
9833:
9810:σ
9781:≥
9720:∞
9717:→
9701:
9689:≠
9664:
9656:∞
9653:→
9614:for each
9584:⋅
9542:∈
9528:⋅
9503:
9477:→
9357:∈
9341:⋅
9299:≥
9218:→
9179:
9173:→
9154:
9071:
9013:
9000:≤
8949:
8806:
8759:
8750:≤
8731:
8621:
8609:≤
8567:
8466:
8460:≤
8445:
8338:
8329:≤
8320:≥
8298:−
8284:
8226:
8217:≤
8208:≥
8199:
8156:is their
8062:∫
8037:
7846:
7831:
7825:≠
7810:
7797:dependent
7771:
7756:
7735:
7656:
7650:⋅
7638:
7603:
7550:∞
7535:∑
7519:
7459:∞
7456:−
7452:∫
7448:−
7423:−
7412:∞
7403:∫
7387:
7328:∞
7323:∞
7320:−
7316:∫
7300:
7185:
7179:≤
7162:
7113:| =
7078:
7057:
7048:
6987:
6916:
6898:
6789:
6760:≥
6731:−
6690:
6684:≤
6672:
6640:
6608:
6579:≤
6530:
6497:∑
6448:∑
6439:
6407:≤
6401:≤
6328:induction
6298:
6270:
6248:
6230:
6202:
6103:⋅
6097:
6079:Linearity
6060:≥
6048:
6022:≥
5954:≥
5923:undefined
5893:γ
5866:−
5852:γ
5844:∞
5839:∞
5836:−
5832:∫
5826:π
5797:γ
5758:∼
5717:≤
5714:α
5698:∞
5685:α
5669:−
5666:α
5658:α
5632:α
5629:−
5619:α
5611:α
5606:∞
5597:∫
5566:α
5549:∼
5485:−
5472:∞
5467:∞
5464:−
5460:∫
5453:π
5405:∼
5374:μ
5347:σ
5343:μ
5340:−
5316:−
5302:∞
5297:∞
5294:−
5290:∫
5277:σ
5273:π
5234:σ
5227:μ
5218:∼
5185:λ
5162:λ
5159:−
5148:λ
5143:∞
5134:∫
5109:λ
5103:
5097:∼
5032:−
5009:∫
4969:∼
4920:−
4906:−
4889:∞
4874:∑
4814:∼
4800:Geometric
4783:λ
4760:λ
4754:λ
4751:−
4735:∞
4720:∑
4695:λ
4681:∼
4636:−
4622:−
4561:∑
4521:∼
4481:⋅
4466:−
4457:⋅
4412:∼
4398:Bernoulli
4389:Notation
4199:∞
4188:−
4177:
4166:∞
4144:
4130:undefined
4119:∞
4108:−
4097:
4086:∞
4064:
4051:∞
4048:−
4038:∞
4027:−
4016:
4005:∞
3983:
3970:∞
3957:∞
3946:−
3935:
3924:∞
3902:
3884:−
3873:
3867:−
3848:
3822:
3724:⋯
3694:⋯
3678:⋅
3659:⋅
3640:⋅
3621:⋅
3589:∞
3574:∑
3558:
3462:∞
3459:−
3455:∫
3451:−
3422:−
3407:∞
3398:∫
3382:
3367:and both
3329:−
3314:∞
3309:μ
3305:∫
3277:μ
3272:∞
3269:−
3265:∫
3241:≤
3235:≤
3215:μ
3212:≥
3187:≤
3181:≤
3170:μ
3167:≤
3104:μ
3084:μ
3069:
2949:∫
2930:Ω
2926:∫
2922:≡
2910:
2809:Borel set
2767:∫
2754:∈
2745:
2674:Ω
2670:∫
2654:
2630:(Ξ©, Ξ£, P)
2523:π
2498:π
2478:
2443:π
2409:∫
2369:∫
2268:∞
2263:∞
2260:−
2256:∫
2240:
2163:
2137:⋯
2084:⋯
1980:∑
1963:
1933:
1894:…
1836:⋅
1665:∞
1650:∑
1634:
1569:$
1566:−
1550:⋅
1544:$
1528:⋅
1522:$
1519:−
1508: bet
1501:$
1489:
1348:⋅
1329:⋅
1310:⋅
1291:⋅
1272:⋅
1253:⋅
1235:
981:⋯
920:
762:¯
635:Notations
602:Etymology
176:Singleton
14069:cumulant
14039:L-moment
14034:kurtosis
14029:skewness
14019:variance
13885:(2019).
13857:(2002).
13799:(1971).
13779:(1968).
13757:(2001).
13723:(1995).
13364:pp. 2β4.
12822:See also
12248:variance
12159:unbiased
10953:(a.s.),
10720:for all
9200:even if
8682:L spaces
8381:variance
6428:we have
5704:if
5681:if
4507:Binomial
4137:if
4057:if
3976:if
3895:if
3799:. Since
2807:for any
2219:integral
1772:Suppose
1767:Examples
1439:roulette
1428:converge
1103:Examples
1085:= β
β
β
=
1029:+ β
β
β
+
611:(a+b)/2.
583:β
257:Variance
13913:3931305
13843:1299979
13809:0270403
13789:0228020
13743:1324786
13066:2309286
12590:vector
12206:moments
12151:samples
12045:is the
11601:(where
9637:Hence,
8720:, then
8379:is the
7289:, then
7281:is the
7237:, then
6976:, then
6887:, then
6855:(a.s.).
6775:(a.s.).
4955:Uniform
4667:Poisson
4270:and so
3779:= βmin(
2593:, with
2205:on the
1608:
1596:
1476:
1464:
1202:
1190:
1097:average
1048:of the
895:, ...,
877:, ...,
865:with a
524:History
488:, with
171:Outcome
13911:
13901:
13865:
13841:
13831:
13807:
13787:
13765:
13741:
13731:
13695:
13348:]
13126:475539
13124:
13064:
13027:
12995:
12866:Median
12855:given
12554:. The
12417:, the
12343:where
12212:; the
12153:, the
11963:where
10872:: Let
10501:: Let
10102:: Let
10065:where
8707:> 1
8700:> 1
8538:|
8406:: Let
8375:where
7585:where
7230:of an
7107:|
6970:(a.s.)
6885:(a.s.)
6749:since
6595:(a.s.)
6120:linear
5744:Cauchy
5535:Pareto
5204:Normal
4415:
4272:E = +β
3768:= max(
3543:, for
1909:where
1703:where
867:finite
749:, and
711:, and
431:moment
429:first
399:, the
118:System
106:Axioms
14065:(pgf)
14054:(mgf)
13978:(cdf)
13972:(pdf)
13966:(pmf)
13350:(PDF)
13344:[
13103:(PDF)
13062:JSTOR
12967:(PDF)
12960:(PDF)
12888:in a
11641:is a
10821:Proof
10675:with
10383:that
9405:with
8903:with
8695:: if
8677:L β L
8545:<
8420:be a
7232:event
6829:then
4356:(see
4354:E = β
4350:E = β
4309:β1)Ο)
2584:ln(2)
2354:) = (
1739:, ...
1718:, ...
1044:as a
869:list
481:, or
427:, or
151:Event
14014:mean
13899:ISBN
13863:ISBN
13829:ISBN
13763:ISBN
13729:ISBN
13693:ISBN
13144:link
13140:link
13122:OCLC
13025:ISBN
12993:ISBN
12693:The
12257:and
12244:size
11093:<
11028:<
10642:Let
10036:for
9977:and
9288:For
8911:and
8899:and
8863:and
8702:and
8424:and
8006:and
7720:are
7700:and
7564:>
7215:Let
6629:and
6142:and
5985:<
5711:<
5688:>
4352:and
4305:6((2
4241:| =
4231:and
4083:<
4035:<
3954:<
3921:<
3803:and
3783:, 0)
3774:and
3772:, 0)
2817:the
2554:and
2552:β ββ
2358:+ Ο)
2311:and
1862:for
1805:and
1724:and
1437:The
1364:3.5.
1162:pips
1116:Let
801:and
695:(in
440:mean
421:mean
13891:doi
13354:doi
13054:doi
13017:doi
12697:in
12465:Var
12413:In
12225:β E
12216:of
12208:of
12137:In
12049:of
11806:If
11367:is
11303:lim
11259:lim
11187:lim
11119:lim
10746:If
10267:lim
9710:lim
9646:lim
8893:β₯ 1
8876:= 2
8718:= 1
8679:of
8520:two
8503:of
8377:Var
8335:Var
8268:βE|
8258:If
7986:of
7680:If
6943:If
6858:If
6118:is
5921:is
5100:exp
4334:6(2
4268:= 0
3763:by
3541:= 2
3529:= 2
2891:of
2825:is
2821:of
2710:of
2577:= β
2566:= β
2559:β β
2221:of
1748:of
1378:die
1224:is
1166:die
1138:die
1038:= 1
851:= E
843:by
823:= E
564:) "
514:or
395:In
14091::
13909:MR
13907:.
13897:.
13853:;
13839:MR
13837:.
13819:;
13805:MR
13785:MR
13753:;
13739:MR
13737:.
13634:^
13583:^
13543:.
13529:^
13282:^
13215:^
13136:}}
13132:{{
13060:.
13050:67
13048:.
13023:.
12934:.
12818:.
12231:.
12197:.
12182:.
12126:.
11851:,
11231:0.
10734:0.
10213:0.
9531:Pr
8925:+
8921:E|
8913:E|
8905:E|
8872:=
8714:+
8415:β
8411::
8026::
7555:Pr
7117:+
6086:)
6063:0.
4338:Ο)
4301:β1
4252:E|
4245:+
3809:+β
3794:β
3790:=
3753:+β
3743:.
3741:+β
3686:16
3675:16
3513:Β±β
3204:or
3010:π
2605:.
2586:.
2475:ln
2160:ln
2072:24
1930:ln
1732:,
1711:,
1605:19
1577:19
1558:38
1547:35
1536:38
1533:37
1473:38
1070:.
853:.
848:ij
839:ij
778:M(
746:av
738:,
707:,
701:E(
598:.
572:.
520:.
477:,
471:E(
462:.
423:,
419:,
415:,
411:,
407:,
13944:e
13937:t
13930:v
13915:.
13893::
13877:)
13871:.
13845:.
13811:.
13791:.
13771:.
13745:.
13715:.
13701:.
13553:.
13362:.
13356::
13173:.
13146:)
13128:.
13105:.
13085:3
13068:.
13056::
13033:.
13019::
13001:.
12976:.
12944:.
12876:m
12861:X
12857:Y
12853:X
12804:2
12790:A
12773:2
12763:A
12753:=
12748:2
12744:)
12740:A
12734:(
12708:A
12681:.
12671:|
12661:A
12654:|
12644:=
12632:A
12599:|
12568:A
12536:.
12531:2
12527:)
12523:]
12520:X
12517:[
12511:E
12508:(
12502:]
12497:2
12493:X
12489:[
12483:E
12480:=
12477:)
12474:X
12471:(
12451:i
12449:p
12444:i
12442:x
12438:.
12436:i
12434:p
12429:i
12427:x
12423:X
12391:.
12386:A
12361:A
12354:1
12331:,
12328:]
12322:A
12315:1
12309:[
12303:E
12300:=
12297:)
12291:A
12283:X
12279:(
12273:P
12223:X
12218:X
12210:X
12202:X
12110:]
12107:)
12104:X
12101:(
12098:g
12095:[
12089:E
12069:.
12066:)
12063:x
12060:(
12057:g
12033:x
12030:d
12024:x
12021:t
12018:i
12011:e
12007:)
12004:x
12001:(
11998:g
11992:R
11983:=
11980:)
11977:t
11974:(
11971:G
11951:,
11948:t
11945:d
11941:)
11938:t
11935:(
11930:X
11922:)
11919:t
11916:(
11913:G
11907:R
11892:2
11888:1
11883:=
11880:]
11877:)
11874:X
11871:(
11868:g
11865:[
11859:E
11835:]
11832:)
11829:X
11826:(
11823:g
11820:[
11814:E
11792:.
11789:x
11786:d
11782:]
11778:t
11775:d
11771:)
11768:t
11765:(
11760:X
11750:x
11747:t
11744:i
11737:e
11730:R
11720:[
11716:)
11713:x
11710:(
11707:g
11701:R
11686:2
11682:1
11677:=
11674:]
11671:)
11668:X
11665:(
11662:g
11659:[
11653:E
11627:R
11617:R
11612::
11609:g
11589:)
11586:X
11583:(
11580:g
11558:.
11555:t
11552:d
11548:)
11545:t
11542:(
11537:X
11527:x
11524:t
11521:i
11514:e
11507:R
11492:2
11488:1
11483:=
11480:)
11477:x
11474:(
11469:X
11465:f
11442:X
11414:X
11392:X
11388:f
11355:}
11350:n
11346:X
11342:{
11322:]
11317:n
11313:X
11307:n
11299:[
11293:E
11290:=
11287:]
11282:n
11278:X
11274:[
11268:E
11263:n
11228:=
11224:|
11220:X
11212:n
11208:X
11203:|
11196:E
11191:n
11165:]
11162:X
11159:[
11153:E
11150:=
11147:]
11142:n
11138:X
11134:[
11128:E
11123:n
11108:;
11090:]
11087:Y
11084:[
11078:E
11071:|
11067:X
11063:|
11056:E
11034:.
11025:]
11022:Y
11019:[
11013:E
10990:+
10984:Y
10977:|
10971:n
10967:X
10962:|
10938:X
10930:n
10926:X
10905:}
10902:0
10896:n
10893::
10888:n
10884:X
10880:{
10852:n
10848:X
10842:n
10834:=
10831:X
10808:.
10805:C
10799:]
10796:X
10793:[
10787:E
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723:ΞΌ
715:X
713:E
709:E
705:)
703:X
682:E
671:E
667:E
649:E
641:E
517:E
501:E
490:E
485:X
483:E
479:E
475:)
473:X
467:X
384:e
377:t
370:v
68:.
50:x
46:e
34:.
20:)
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