Law of large numbers

580: 57: 10504:. "We don't believe in such laws as laws of large numbers. This is sort of, uh, old dogma, I think, that was cooked up by somebody " said Tim Cook and while: "However, the law of large numbers has nothing to do with large companies, large revenues, or large growth rates. The law of large numbers is a fundamental concept in probability theory and statistics, tying together theoretical probabilities that we can calculate to the actual outcomes of experiments that we empirically perform. 362: 1485: 1467: 4807: 1420: 3997:, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event is expected to occur approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if 5181: 8277:, one could easily obtain the probability mass function. For each event in the objective probability mass function, one could approximate the probability of the event's occurrence with the proportion of times that any specified event occurs. The larger the number of repetitions, the better the approximation. As for the continuous case: 6704: 4588: 1201: 3974: 738: 5017: 1727:

as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of the average

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With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic

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in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will

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Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability).

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As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in the series, keeping the expected value constant. If the variances are bounded, then the law applies, as

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Simulation illustrating the law of large numbers. Each frame, a coin that is red on one side and blue on the other is flipped, and a dot is added in the corresponding column. A pie chart shows the proportion of red and blue so far. Notice that while the proportion varies significantly at first, it

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The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed,

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This image illustrates the convergence of relative frequencies to their theoretical probabilities. The probability of picking a red ball from a sack is 0.4 and black ball is 0.6. The left plot shows the relative frequency of picking a black ball, and the right plot shows the relative frequency of

5574: 373:. As the number of rolls in this run increases, the average of the values of all the results approaches 3.5. Although each run would show a distinctive shape over a small number of throws (at the left), over a large number of rolls (to the right) the shapes would be extremely similar. 3212: 5008: 5837: 2799: 8206: 848:. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the 1970: 6567: 5956: 3033: 6043: 5272: 4485: 2198: 1595: 3829: 8469: 783:(1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by 428:

wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a

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goes to infinity, the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate subsequence.

4802:{\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.} 4094:

This theorem makes rigorous the intuitive notion of probability as the expected long-run relative frequency of an event's occurrence. It is a special case of any of several more general laws of large numbers in probability theory.

1415:{\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.} 673:

to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.

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exists for the weak law of large numbers to be true. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of

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of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the

1074: 5635: 4402: 2649: 8062: 7330: 7207: 5176:{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|<\varepsilon )=1-\operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\geq 1-{\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.} 1004: 1789:, which is not bounded. At each stage, the average will be normally distributed (as the average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is 433:

of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the

4090: 5393: 6208: 4212: 1899: 8683: 5891: 2873: 8830: 7439: 6954: 2565: 2431: 7930: 4313: 1476: 6888: 4561: 1174: 1719:), no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin. 584:

picking a red ball, both over 10,000 trials. As the number of trials increases, the relative frequencies approach their respective theoretical probabilities, demonstrating the Law of Large Numbers.

5972: 5201: 4414: 2107: 1524: 7675: 8332: 8962: 6265: 2487: 6150: 1622: 6735: 4863: 545: 3509: 8037: 3456: 3274: 6472: 3406: 9198: 9139: 9079: 9019: 3739: 2096:(this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that). 702:. The Cauchy distribution and the Pareto distribution represent two cases: the Cauchy distribution does not have an expectation, whereas the expectation of the Pareto distribution ( 8882: 2080:. This view justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average". 7839: 6699:{\displaystyle \lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\iff \exists \epsilon >0,\left|{\frac {S_{n}(\omega )}{n}}\right|\geq \epsilon \ {\mbox{infinitely often}},} 6380: 5881: 5619: 4903: 3346: 2864: 2371: 729:, typical in human economic/rational behaviour, the law of large numbers does not help in solving the bias. Even if the number of trials is increased the selection bias remains. 2636: 8743: 3658:), ...} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E. This is the 8252: 7743: 7544: 7484: 6730: 2230: 464:

Throughout its history, many mathematicians have refined this law. Today, the LLN is used in many fields including statistics, probability theory, economics, and insurance.

1832: 1787: 8325: 6990: 3458:, then the average at any point will also be normally distributed. The width of the distribution of the average will tend toward zero (standard deviation asymptotic to 1993: 1866: 7781: 7976: 6301: 6294: 7571: 6094: 3969:{\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} \right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.} 1016: 6097: 1509: 898: 573: 8226: 8057: 7950: 7698: 4581: 1194: 4320: 6998: 927: 4027: 748:

molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container.

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on . Using traditional methods to compute this integral is very difficult, so the Monte Carlo Method can be used here. Using the above algorithm, we get

5569:{\displaystyle \varphi _{{\frac {1}{n}}X}(t)=\varphi _{X}({\tfrac {t}{n}})\quad {\text{and}}\quad \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)\quad } 5316: 8489:, which uses a random sampling of numbers to approximate numerical results. The algorithm to compute an integral of f(x) on an interval is as follows: 4140: 1449:

The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see

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This shows that the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists.

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by independence, and the last term is zero --- and similarly for the other terms. Therefore the only terms in the sum with nonzero expectation are

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There are extensions of the law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name

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With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations.

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on . Using the Monte Carlo Method and the LLN, we can see that as the number of samples increases, the numerical value gets closer to 0.4180233.

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The law of large numbers provides an expectation of an unknown distribution from a realization of the sequence, but also any feature of the

5304: 3207:{\displaystyle {\begin{cases}1-F(x)&={\frac {e}{2x\ln(x)}},&x\geq e\\F(x)&={\frac {e}{-2x\ln(-x)}},&x\leq -e\end{cases}}} 549:

According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the

17: 10044: 6155: 5003:{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.} 718:

such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as

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of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the

5832:{\displaystyle \varphi _{{\overline {X}}_{n}}(t)=\left^{n}=\left^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\quad n\to \infty .} 592:

toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to

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The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of

4816: 2794:{\displaystyle E\left({\frac {\sin(X)e^{X}}{X}}\right)=\ \int _{x=0}^{\infty }{\frac {\sin(x)e^{x}}{x}}e^{-x}dx={\frac {\pi }{2}}} 602:. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 485: 10533: 8748: 7335: 821:("the law of large numbers"). Thereafter, it was known under both names, but the "law of large numbers" is most frequently used. 10004: 9610:

Probabilité des jugements en matière criminelle et en matière civile, précédées des règles générales du calcul des probabilitiés

8201:{\displaystyle \Pr(|S_{n}|\geq n\epsilon )\leq {\frac {1}{(n\epsilon )^{4}}}{\mathbb {E} }\leq {\frac {C}{\epsilon ^{4}n^{2}}},} 2514: 2380: 4242: 824:

After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including

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Kroese, Dirk P.; Brereton, Tim; Taimre, Thomas; Botev, Zdravko I. (2014). "Why the Monte Carlo method is so important today".

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has no expected value according to Lebesgue integration, but using conditional convergence and interpreting the integral as a

10364: 10257: 9773: 9440: 5960:μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see 4514: 1127: 9084:

We observe that as n increases, the numerical value also increases. When we get the actual results for the integral we get

1965:{\displaystyle {\overline {X}}_{n}\ {\overset {\text{a.s.}}{\longrightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 857:

of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.

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With this method, one can cover the whole x-axis with a grid (with grid size 2h) and obtain a bar graph which is called a

7576: 5951:{\displaystyle {\overline {X}}_{n}\,{\overset {\mathcal {D}}{\rightarrow }}\,\mu \qquad {\text{for}}\qquad n\to \infty .} 3028:{\displaystyle E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{x=1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)} 8890: 6213: 2436: 10323: 10291: 9987: 9506: 9473: 9302: 9209: 6103: 472:

For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal

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and this is unbounded. If we replace the random variables with Gaussian variables having the same variances, namely

9379: 3038: 1196:) and no correlation between random variables. In that case, the variance of the average of n random variables is 10538: 8274: 5961: 1450: 212: 148: 9144:

When the LLN was used, the approximation of the integral was closer to its true value, and thus more accurate.

7981: 3461: 260: 121: 3411: 3229: 10523: 10443: 9234: 6038:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 5267:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 4480:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 3725: 3359: 2193:{\displaystyle {\overline {X}}_{n}-\operatorname {E} {\big }\ {\overset {\text{a.s.}}{\longrightarrow }}\ 0,} 1590:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 1443: 787:. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his 706:<1) is infinite. One way to generate the Cauchy-distributed example is where the random numbers equal the 9792: 9153: 9089: 9029: 8969: 10528: 10244:, Lecture Notes in Physics, vol. 739, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 63–78, 9923: 8835: 10433: 9588:

Ars Conjectandi: Usum & Applicationem Praecedentis Doctrinae in Civilibus, Moralibus & Oeconomicis

8464:{\displaystyle {\frac {N_{n}(C)}{n}}\thickapprox p=P(X\in C)=\int _{a-h}^{a+h}f(x)\,dx\thickapprox 2hf(a)} 7786: 655:

approach zero. Intuitively, the expected difference grows, but at a slower rate than the number of flips.

10438: 9259: 5883: 5852: 5590: 4874: 3302: 2810: 2342: 2586: 1706:{\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=1.} 9734: 9249: 9239: 5190: 1513: 413: 399: 116: 28: 10501: 8688: 5846: 3527:. Since the width of the distribution of the average is not zero, it must have a positive lower bound 9224: 6812:{\displaystyle \Pr \left(\omega :|S_{n}(\omega )|\geq n\epsilon {\mbox{ infinitely often}}\right)=0.} 4868: 4099: 699: 565: 232: 9738: 8231: 7703: 7489: 7444: 6709: 3053: 10492: 8270: 8261:

For a proof without the added assumption of a finite fourth moment, see Section 22 of Billingsley.

2580: 837: 804: 291: 286: 175: 160: 4583:). The independence of the random variables implies no correlation between them, and we have that 1796: 1743: 9613: 9219: 8280: 6557:{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\right)=0,} 1888: 270: 141: 6959: 2077: 1871:

There are also examples of the weak law applying even though the expected value does not exist.

812: 10543: 9465: 9294: 3810:{\displaystyle \left\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .} 3613: 2804: 165: 9977: 9498: 1837: 10474: 10051: 9696: 9214: 7748: 6462:{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}=0\right)=1.} 1737: 1439: 1111: 579: 557: 409:

The LLN is important because it guarantees stable long-term results for the averages of some

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goes to infinity. As an example, assume that each random variable in the series follows a

1432:. Large or infinite variance will make the convergence slower, but the LLN holds anyway. 8: 5296: 854: 741: 691: 687: 651: 620: 255: 197: 185: 180: 9941: 2433:

happens an infinite number of times, although at infrequent intervals. (Not necessarily

10413: 10395: 10237: 10200: 10159: 10076: 9836: 9684: 9645: 9553: 9413: 9343: 8486: 8211: 8042: 7935: 7683: 4566: 3979: 2639: 2306:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} <\infty .} 1505: 1179: 845: 659: 378: 242: 131: 71: 48: 3348:

Kolmogorov's strong law does not apply because the partial sum in his criterion up to

10471: 10452: 10360: 10319: 10287: 10253: 10204: 10019: 9983: 9877: 9840: 9769: 9688: 9665:"Démonstration élémentaire d'une proposition générale de la théorie des probabilités" 9545: 9502: 9491: 9469: 9458: 9436: 9335: 9298: 9287: 8228:

sufficiently large, and therefore this series is summable. Since this holds for any

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Interpreting this result, the weak law states that for any nonzero margin specified (

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Yao, Kai; Gao, Jinwu (2016). "Law of Large Numbers for Uncertain Random Variables".

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We give a relatively simple proof of the strong law under the assumptions that the

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is the characteristic function of the constant random variable μ, and hence by the

5300: 2643: 1115: 808: 780: 568:, the expected value is the theoretical probability of success, and the average of 126: 56: 2059:{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.} 10343: 9229: 8517:

independent and identically distributed (i.i.d.) random variables on . Then let X

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One application of the LLN is the important method of approximation known as the

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which can be used to shorten and simplify the proofs. This assumption of finite

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where all subscripts are distinct, must have zero expectation. This is because

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This result is useful to derive consistency of a large class of estimators (see

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will not occur. It does not imply that with probability 1, we have that for any

799:) in 1713. He named this his "Golden Theorem" but it became generally known as " 10496: 10350:. Handbook of econometrics. Vol. IV. Elsevier Science. pp. 2111–2245. 9331: 6370:{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=0\right)=1,} 4112: 2866:

does not have an expected value in the conventional sense because the infinite

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A modern introduction to probability and statistics: understanding why and how

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which can be done using a software, and use a random number table that gives U

4503: 3994: 2571:, since the convergence is not necessarily uniform on the set where it holds. 1475: 1069:{\displaystyle {\overline {X}}_{n}\to \mu \quad {\textrm {as}}\ n\to \infty .} 833: 10517: 10383: 9881: 9872: 9855: 9680: 9549: 9339: 2870:

is not absolutely convergent, but using conditional convergence, we can say:

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The strong law does not hold in the following cases, but the weak law does.

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The strong law of large numbers can itself be seen as a special case of the

553:) will approach 3.5, with the precision increasing as more dice are rolled. 10005:"A Note on the Weak Law of Large Numbers for Exchangeable Random Variables" 9244: 5411:, ... have the same characteristic function, so we will simply denote this 366: 317: 227: 111: 9628:

Hacking, Ian (1983). "19th-century Cracks in the Concept of Determinism".

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Introductory probability texts often additionally assume identical finite

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is zero, but the expected value does not exist, and indeed the average of

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are independent and identically distributed uniform random variables on .

4397:{\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).} 4126: 663: 550: 473: 237: 78: 66: 2085: 369:

of the law of large numbers using a particular run of rolls of a single

10409: 10195: 10178: 10163: 10139: 9831: 9814: 9649: 9557: 7325:{\displaystyle X_{i}^{3}X_{j},X_{i}^{2}X_{j}X_{k},X_{i}X_{j}X_{k}X_{l}} 7202:{\displaystyle {\mathbb {E} }={\mathbb {E} }\left={\mathbb {E} }\left.} 5189:

approaches infinity, the expression approaches 1. And by definition of

1790: 1512:(iid) samples from a random variable with finite mean, the sample mean 999:{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})} 410: 95: 41: 10502:

Apple CEO Tim Cook said something that would make statisticians cringe

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He attempts a two-part proof of the law on pp. 139–143 and pp. 277 ff.

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then the Borel-Cantelli Lemma implies the result. So let us estimate

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distributed random variable with probability 0.5. The random variable

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If the summands are independent but not identically distributed, then

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What this means is that the probability that, as the number of trials

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are identically distributed, all of these are the same, and moreover

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Let us first note that without loss of generality we can assume that

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The common mean μ of the sequence is the mean of the sample average:

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Reiter, Detlev (2008), Fehske, H.; Schneider, R.; Weiße, A. (eds.),

10155: 10140:"An Analytic Technique to Prove Borel's Strong Law of Large Numbers" 9664: 9641: 5587:

These rules can be used to calculate the characteristic function of

4085:{\displaystyle {\frac {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .} 361: 5388:{\displaystyle \varphi _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.} 4509: 4119: 1425: 1122: 425: 222: 10450: 10400: 6203:{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}<\infty } 4207:{\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.} 2583:

distributed random variable with parameter 1. The random variable

1106:) exists according to Lebesgue integration and is finite. It does 9760:. Springer Texts in Statistics. New York, NY: Springer New York. 8678:{\displaystyle (b-a){\tfrac {f(X_{1})+f(X_{2})+...+f(X_{n})}{n}}} 5423:

Among the basic properties of characteristic functions there are

1740:(normal distribution) with mean zero, but with variance equal to 707: 394: 10183:

Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete

9435:. Springer texts in statistics. London : Springer. p. 187. 6706:

and thus to prove the strong law we need to show that for every

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is an example of the law of large numbers. Initially, there are

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and then by the Strong Law of Large Numbers, this converges to

4236: 3583:)=1 and the average will attain ε an infinite number of times.) 745: 711: 417: 10386:(2013). "A Tricentenary history of the Law of Large Numbers". 10107:"Asymptotic Properties of Non-Linear Least Squares Estimators" 9969: 9756:

Bhattacharya, Rabi; Lin, Lizhen; Patrangenaru, Victor (2016).

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With a single molecule, the motion appears to be quite random.

4315:, we are interested in the convergence of the sample average 710:

of an angle uniformly distributed between −90° and +90°. The

421: 9942:"What Is the Law of Large Numbers? (Definition) | Built In" 9758:

A Course in Mathematical Statistics and Large Sample Theory

9755: 8825:{\displaystyle (b-a)\int _{a}^{b}f(x){\tfrac {1}{b-a}}{dx}} 7932:

Note that the right-hand side is a quadratic polynomial in

7434:{\displaystyle {\mathbb {E} }={\mathbb {E} }{\mathbb {E} }} 6949:{\displaystyle \sum _{n=1}^{\infty }\Pr(A_{n})<\infty ,} 4504:

Proof using Chebyshev's inequality assuming finite variance

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states the conditions under which the convergence happens

2560:{\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } 2426:{\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon } 34:

Averages of repeated trials converge to the expected value

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Wiley Interdisciplinary Reviews: Computational Statistics

9395: 7925:{\displaystyle {\mathbb {E} }=n\tau +3n(n-1)\sigma ^{4}.} 4308:{\displaystyle E(X_{1})=E(X_{2})=\cdots =\mu <\infty } 1834:. The variance of the average is therefore asymptotic to 922:, both versions of the law state that the sample average 10318:] (in Danish) (3rd ed.). Copenhagen: HCØ-tryk. 10179:"An elementary proof of the strong law of large numbers" 9815:"An elementary proof of the strong law of large numbers" 6883:{\displaystyle A_{n}=\{\omega :|S_{n}|\geq n\epsilon \}} 3214:

then it has no expected value, but the weak law is true.

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Although the proportion of heads (and tails) approaches

4556:{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} 1169:{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} 8795: 8590: 6792: 6687: 5478: 4634: 4345: 3515:, there is probability which does not go to zero with 3464: 3414: 3280:

so that the denominator is positive) with probability

3232: 1247: 9714: 9712: 9156: 9092: 9032: 8972: 8893: 8838: 8751: 8691: 8573: 8335: 8283: 8234: 8214: 8065: 8045: 7984: 7958: 7938: 7847: 7789: 7751: 7706: 7686: 7579: 7552: 7492: 7447: 7338: 7215: 7001: 6962: 6896: 6825: 6738: 6712: 6570: 6475: 6383: 6304: 6296:

by centering. In this case, the strong law says that

6276: 6216: 6158: 6106: 6075: 5975: 5894: 5855: 5638: 5593: 5431: 5319: 5204: 5020: 4913: 4877: 4819: 4591: 4569: 4517: 4417: 4323: 4245: 4143: 4030: 3832: 3742: 3362: 3305: 3047: 2876: 2813: 2652: 2589: 2517: 2439: 2383: 2345: 2233: 2110: 1996: 1902: 1840: 1799: 1746: 1625: 1527: 1204: 1182: 1130: 1019: 930: 488: 3575:) that it will happen. (This seems to indicate that 564:

will converge to the theoretical probability. For a

9976:Lehmann, Erich L.; Romano, Joseph P. (2006-03-30). 9493:

A Modern Introduction to Probability and Statistics

9460:

A Modern Introduction to Probability and Statistics

9289:

A Modern Introduction to Probability and Statistics

7670:{\displaystyle {\mathbb {E} }=({\mathbb {E} })^{2}} 5291:

Proof using convergence of characteristic functions

2327:

Differences between the weak law and the strong law

2086:

differences between the weak law and the strong law

9709: 9192: 9133: 9073: 9013: 8956: 8876: 8824: 8737: 8677: 8463: 8319: 8246: 8220: 8200: 8051: 8031: 7970: 7944: 7924: 7833: 7775: 7737: 7692: 7669: 7565: 7538: 7478: 7433: 7324: 7201: 6984: 6948: 6882: 6811: 6724: 6698: 6556: 6461: 6369: 6288: 6259: 6202: 6144: 6088: 6037: 5950: 5875: 5831: 5613: 5568: 5387: 5266: 5175: 5002: 4897: 4857: 4801: 4575: 4555: 4479: 4396: 4307: 4206: 4084: 3968: 3809: 3535:), which means there is a probability of at least 3503: 3450: 3400: 3340: 3268: 3206: 3027: 2858: 2793: 2630: 2559: 2481: 2425: 2365: 2305: 2192: 2092:then for the average to converge almost surely on 2058: 1964: 1860: 1826: 1781: 1705: 1589: 1414: 1188: 1168: 1068: 998: 556:It follows from the law of large numbers that the 539: 10281: 10012:Communications of the Korean Mathematical Society 9524:"Cauchy-Distributed Functions of Cauchy Variates" 6308: 2000: 1645: 1494:approaches 50% as the number of trials increases. 10515: 8957:{\displaystyle f(x)=cos^{2}(x){\sqrt {x^{3}+1}}} 8066: 6963: 6918: 6739: 6572: 6491: 6476: 6399: 6384: 6315: 6305: 6260:{\displaystyle {\mathbb {E} }=:\tau <\infty } 4144: 3834: 3588: 2482:{\displaystyle |{\overline {X}}_{n}-\mu |\neq 0} 2007: 1997: 1642: 1627: 1110:mean that the associated probability measure is 899:independent and identically distributed (i.i.d.) 574:independent and identically distributed (i.i.d.) 10309: 9669:Journal für die reine und angewandte Mathematik 9612:(in French). Paris, France: Bachelier. p. 9380:"The Law of Large Numbers and its Applications" 6145:{\displaystyle {\mathbb {E} }=:\mu <\infty } 3985: 612:. In particular, the proportion of heads after 453:results gets close to the expected value times 10348:Large sample estimation and hypothesis testing 9793:"The strong law of large numbers – What's new" 9701:: CS1 maint: DOI inactive as of August 2024 ( 9521: 10341: 10092: 9975: 2163: 2139: 869:that are described below. They are called the 342: 9601: 6877: 6839: 4858:{\displaystyle E({\overline {X}}_{n})=\mu .} 4239:random variables with finite expected value 2377:. Thus, it leaves open the possibility that 816: 540:{\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5} 10217: 9917: 9915: 8327:, for small positive h. Thus, for large n: 7209:We first claim that every term of the form 3567:, there is still a probability of at least 1438:of the random variables can be replaced by 788: 10282:Grimmett, G. R.; Stirzaker, D. R. (1992). 9921: 9787: 9785: 9729: 9727: 9662: 9600:Poisson names the "law of large numbers" ( 9522:Pitman, E. J. G.; Williams, E. J. (1967). 6625: 6621: 6064: 5012:This may be used to obtain the following: 3504:{\textstyle 1/{\sqrt {2\log \log \log n}}} 1508:'s law) states that given a collection of 349: 335: 10399: 10373: 10286:(2nd ed.). Oxford: Clarendon Press. 10238:"The Monte Carlo Method, an Introduction" 10194: 10122: 9871: 9830: 9733: 9590:(in Latin). Translated by Sheynin, Oscar. 9585: 9539: 8433: 8137: 8032:{\displaystyle {\mathbb {E} }\leq Cn^{2}} 7987: 7850: 7795: 7709: 7631: 7582: 7495: 7450: 7410: 7382: 7341: 7101: 7035: 7004: 6219: 6109: 5925: 5912: 5790: 5786: 4508:This proof uses the assumption of finite 3451:{\textstyle {\sqrt {k/\log \log \log k}}} 3269:{\textstyle {\sqrt {k/\log \log \log k}}} 2320: 1691: 1651: 10104: 9912: 9570: 5311:, with finite mean μ, can be written as 4216: 3547:trials. It will happen with probability 865:There are two different versions of the 772:), despite its underlying random nature. 736: 578: 360: 10489:Animations for the Law of Large Numbers 10300: 10176: 9853: 9819:Wahrscheinlichkeitstheorie Verw Gebiete 9812: 9782: 9724: 9627: 9607: 9488: 9455: 9430: 9284: 3543:) that the average will attain ε after 3519:, while the average sometime after the 3401:{\displaystyle \log n/\log \log \log n} 1887:'s law) states that the sample average 1510:independent and identically distributed 904:random variables with expected value E( 807:, named after Jacob Bernoulli's nephew 658:Another good example of the LLN is the 576:) is precisely the relative frequency. 14: 10516: 10382: 10235: 9795:. Terrytao.wordpress.com. 19 June 2008 9718: 9317: 9193:{\displaystyle {\frac {e^{x}-1}{e-1}}} 9147:Another example is the integration of 9134:{\displaystyle \int _{-1}^{2}f(x){dx}} 9074:{\displaystyle \int _{-1}^{2}f(x){dx}} 9014:{\displaystyle \int _{-1}^{2}f(x){dx}} 8254:, we have established the Strong LLN. 4406:The weak law of large numbers states: 10470: 10451: 10332: 10231: 10229: 10111:The Annals of Mathematical Statistics 10042: 9906: 9894: 9575:. New York: Random House. p. 50. 9528:The Annals of Mathematical Statistics 8877:{\displaystyle \int _{a}^{b}f(x){dx}} 662:. These methods are a broad class of 10354: 10074: 10003:Dguvl Hun Hong; Sung Ho Lee (1998). 9964: 9373: 9371: 9369: 9367: 9365: 9363: 9361: 9359: 9357: 9280: 9278: 9276: 9274: 8258:Another proof was given by Etemadi. 7834:{\displaystyle ({\mathbb {E} })^{2}} 5966: 5195: 4408: 2101: 1893: 1518: 1010: 815:further described it under the name 803:". This should not be confused with 10242:Computational Many-Particle Physics 10137: 8493:Simulate uniform random variables X 5876:{\displaystyle {\overline {X}}_{n}} 5614:{\displaystyle {\overline {X}}_{n}} 4898:{\displaystyle {\overline {X}}_{n}} 4024:trials, then with probability one, 4005:its probability of occurrence, and 3341:{\displaystyle k/\log \log \log k.} 2859:{\displaystyle 2^{X}(-1)^{X}X^{-1}} 2366:{\displaystyle {\overline {X}}_{n}} 24: 10376:Large sample methods in statistics 10312:Videregående Sandsynlighedsregning 10226: 9320:IEEE Transactions on Fuzzy Systems 6940: 6913: 6626: 6582: 6501: 6409: 6325: 6254: 6197: 6139: 6029: 5942: 5919: 5823: 5258: 5081: 5021: 4914: 4471: 4302: 4076: 3954: 3913: 3844: 3801: 2948: 2720: 2631:{\displaystyle \sin(X)e^{X}X^{-1}} 2335:states that for a specified large 2297: 2250: 2131: 2017: 1956: 1637: 1581: 1060: 572:such variables (assuming they are 25: 10555: 10426: 10374:Sen, P. K; Singer, J. M. (1993). 10144:The American Mathematical Monthly 9377: 9354: 9271: 9210:Asymptotic equipartition property 8440: 8364: 1613:That is, for any positive number 1008:converges to the expected value: 897:, ... is an infinite sequence of 10303:Probability: Theory and Examples 10284:Probability and Random Processes 9854:Kingman, J. F. C. (April 1978). 8738:{\displaystyle (b-a)E(f(X_{1}))} 8059:sufficiently large. By Markov, 3276:(starting at sufficiently large 3039:cumulative distribution function 1483: 1474: 1465: 1099:means that the expected value E( 480:of the average of the rolls is: 55: 10359:(8th ed.). Prentice Hall. 10211: 10170: 10131: 10098: 10086: 10068: 10036: 9996: 9958: 9934: 9900: 9888: 9847: 9806: 9749: 9656: 9630:Journal of the History of Ideas 9621: 9594: 9579: 9225:Keynes' Treatise on Probability 8480: 8264: 6012: 5962:Convergence of random variables 5935: 5929: 5816: 5810: 5565: 5498: 5492: 5375: 5241: 4454: 4001:denotes the event in question, 3786: 2225:has a finite second moment and 1939: 1564: 1451:Convergence of random variables 1043: 10534:Asymptotic theory (statistics) 10305:(2nd ed.). Duxbury Press. 9979:Weak law converges to constant 9586:Bernoulli, Jakob (1713). "4". 9564: 9515: 9482: 9449: 9424: 9389: 9311: 9120: 9114: 9060: 9054: 9000: 8994: 8931: 8925: 8903: 8897: 8863: 8857: 8791: 8785: 8764: 8752: 8732: 8729: 8716: 8710: 8704: 8692: 8665: 8652: 8631: 8618: 8609: 8596: 8586: 8574: 8458: 8452: 8430: 8424: 8388: 8376: 8355: 8349: 8314: 8290: 8247:{\displaystyle \epsilon >0} 8160: 8142: 8123: 8113: 8101: 8088: 8073: 8069: 8010: 7992: 7906: 7894: 7873: 7855: 7822: 7818: 7800: 7790: 7770: 7758: 7738:{\displaystyle {\mathbb {E} }} 7732: 7714: 7658: 7654: 7636: 7626: 7620: 7587: 7539:{\displaystyle {\mathbb {E} }} 7533: 7500: 7479:{\displaystyle {\mathbb {E} }} 7473: 7455: 7428: 7415: 7405: 7387: 7374: 7346: 7027: 7009: 6979: 6966: 6934: 6921: 6864: 6849: 6778: 6774: 6768: 6754: 6725:{\displaystyle \epsilon >0} 6664: 6658: 6622: 6606: 6600: 6579: 6525: 6519: 6498: 6469:It is equivalent to show that 6433: 6427: 6406: 6322: 6242: 6224: 6178: 6165: 6127: 6114: 6026: 5998: 5939: 5915: 5820: 5787: 5669: 5663: 5562: 5556: 5543: 5537: 5521: 5515: 5489: 5474: 5458: 5452: 5379: 5369: 5363: 5336: 5330: 5255: 5227: 5129: 5087: 5069: 5027: 4962: 4920: 4843: 4823: 4741: 4709: 4680: 4677: 4645: 4630: 4618: 4598: 4537: 4524: 4468: 4440: 4388: 4356: 4284: 4271: 4262: 4249: 4235:, ... an infinite sequence of 4178: 4165: 4151: 4147: 4073: 4059: 4050: 4044: 3950: 3944: 3940: 3937: 3925: 3919: 3907: 3888: 3850: 3783: 3777: 3767: 3763: 3751: 3744: 3713:s, and measurable function of 3523:th trial will come back up to 3174: 3165: 3136: 3130: 3103: 3097: 3071: 3065: 3022: 3016: 2976: 2966: 2907: 2897: 2834: 2824: 2740: 2734: 2675: 2669: 2602: 2596: 2547: 2519: 2469: 2441: 2413: 2385: 2291: 2278: 2173: 2014: 1953: 1925: 1776: 1764: 1681: 1653: 1634: 1578: 1550: 1354: 1322: 1293: 1290: 1258: 1243: 1231: 1211: 1150: 1137: 1057: 1037: 993: 961: 698:becomes larger; the reason is 694:(α<1) will not converge as 122:Collectively exhaustive events 13: 1: 10475:"Strong Law of Large Numbers" 10357:A first course in probability 10274: 10218:Billingsley, Patrick (1979). 9924:"Strong law of large numbers" 9431:Dekking, Michel, ed. (2005). 9235:Law of the iterated logarithm 7952:, and as such there exists a 3589:Uniform laws of large numbers 1874: 1446:in both versions of the law. 677: 10105:Jennrich, Robert I. (1969). 10018:(2): 385–391. Archived from 8887:We can find the integral of 8275:Borel's law of large numbers 6336: 5982: 5901: 5862: 5650: 5600: 5211: 5101: 5041: 4934: 4884: 4832: 4607: 4424: 4330: 3991:Borel's law of large numbers 3986:Borel's law of large numbers 3671:uniform law of large numbers 3595:uniform law of large numbers 2529: 2451: 2395: 2352: 2150: 2117: 2028: 1909: 1827:{\displaystyle n^{2}/\log n} 1782:{\displaystyle 2n/\log(n+1)} 1663: 1534: 1220: 1026: 937: 883:. Stated for the case where 441:The LLN only applies to the 7: 10456:"Weak Law of Large Numbers" 10439:Encyclopedia of Mathematics 10316:Advanced Probability Theory 10250:10.1007/978-3-540-74686-7_3 10177:Etemadi, Nasrollah (1981). 9928:Encyclopedia of Mathematics 9743:Encyclopedia of Mathematics 9260:Strong law of small numbers 9203: 8525:for i= 1, 2, ..., n. Then X 8320:{\displaystyle C=(a-h,a+h]} 6890:, and if we can show that 6051: 5280: 4493: 2567:holds for all large enough 2315:This statement is known as 2206: 1978: 1881:strong law of large numbers 1603: 1456: 1092:(Lebesgue integrability of 1082: 818:"la loi des grands nombres" 467: 18:Strong law of large numbers 10: 10560: 9332:10.1109/TFUZZ.2015.2466080 9250:Regression toward the mean 9240:Law of truly large numbers 6985:{\displaystyle \Pr(A_{n})} 5191:convergence in probability 3736:) such that E < ∞, and 3669:A particular example of a 3290:for each. The variance of 779:The Italian mathematician 732: 725:And if the trials embed a 560:of success in a series of 29:Law of truly large numbers 26: 10337:(4th ed.). Springer. 10301:Durrett, Richard (1995). 10093:Newey & McFadden 1994 9860:The Annals of Probability 9856:"Uses of Exchangeability" 9766:10.1007/978-1-4939-4032-5 9603:la loi des grands nombres 5884:converges in distribution 2078:pointwise ergodic theorem 1502:weak law of large numbers 566:Bernoulli random variable 10310:Martin Jacobsen (1992). 9681:10.1515/crll.1846.33.259 9489:Dekking, Michel (2005). 9456:Dekking, Michel (2005). 9285:Dekking, Michel (2005). 9265: 8271:probability distribution 5307:of any random variable, 3820:Then E is continuous in 3702:) is continuous at each 3041:of a random variable is 1861:{\displaystyle 1/\log n} 1514:converges in probability 860: 424:in a single spin of the 292:Law of total probability 287:Conditional independence 176:Exponential distribution 161:Probability distribution 27:Not to be confused with 10491:by Yihui Xie using the 10220:Probability and Measure 10124:10.1214/aoms/1177697731 9813:Etemadi, N. Z. (1981). 9683:(inactive 2024-08-02). 9663:Tchebichef, P. (1846). 9608:Poisson, S. D. (1837). 9541:10.1214/aoms/1177698885 9220:Infinite monkey theorem 8555:Take the average of f(X 7776:{\displaystyle 3n(n-1)} 6065:Proof of the strong law 5847:Lévy continuity theorem 5305:characteristic function 3620:∈ Θ, and continuous in 2642:, which is an improper 2317:Kolmogorov's strong law 1889:converges almost surely 1732:values goes to zero as 797:The Art of Conjecturing 686:results taken from the 416:. For example, while a 271:Conditional probability 10539:Theorems in statistics 10434:"Law of large numbers" 10355:Ross, Sheldon (2009). 10333:Loève, Michel (1977). 10077:"Law of large numbers" 10045:"Law of large numbers" 9909:, Chapter 17.3, p. 251 9873:10.1214/aop/1176995566 9739:"Law of large numbers" 9602: 9194: 9135: 9075: 9015: 8958: 8878: 8826: 8739: 8679: 8465: 8321: 8248: 8222: 8202: 8053: 8033: 7972: 7971:{\displaystyle C>0} 7946: 7926: 7835: 7777: 7739: 7694: 7671: 7567: 7540: 7480: 7435: 7326: 7203: 7070: 6986: 6950: 6917: 6884: 6813: 6794: infinitely often 6726: 6700: 6558: 6463: 6371: 6290: 6289:{\displaystyle \mu =0} 6261: 6204: 6146: 6090: 6039: 5952: 5877: 5833: 5615: 5570: 5389: 5268: 5177: 5004: 4899: 4869:Chebyshev's inequality 4859: 4803: 4577: 4557: 4481: 4398: 4309: 4208: 4100:Chebyshev's inequality 4086: 4016:) the number of times 3970: 3884: 3811: 3505: 3452: 3402: 3342: 3270: 3208: 3029: 2952: 2860: 2795: 2632: 2561: 2483: 2427: 2367: 2321:Sen & Singer (1993 2307: 2254: 2194: 2060: 1966: 1891:to the expected value 1862: 1828: 1783: 1707: 1591: 1516:to the expected value 1416: 1190: 1170: 1070: 1000: 817: 789: 776: 669:that rely on repeated 585: 541: 402:converges to the true 374: 213:Continuous or discrete 166:Bernoulli distribution 10378:. Chapman & Hall. 10189:. Springer: 119–122. 9571:Mlodinow, L. (2008). 9497:. Springer. pp. 9464:. Springer. pp. 9293:. Springer. pp. 9215:Central limit theorem 9195: 9136: 9076: 9016: 8959: 8879: 8827: 8740: 8680: 8466: 8322: 8249: 8223: 8203: 8054: 8034: 7973: 7947: 7927: 7836: 7778: 7740: 7695: 7672: 7568: 7566:{\displaystyle X_{i}} 7541: 7481: 7436: 7327: 7204: 7050: 6987: 6951: 6897: 6885: 6814: 6727: 6701: 6559: 6464: 6372: 6291: 6262: 6205: 6147: 6091: 6089:{\displaystyle X_{i}} 6040: 5953: 5878: 5834: 5616: 5571: 5390: 5269: 5178: 5005: 4900: 4860: 4804: 4578: 4558: 4482: 4399: 4310: 4217:Proof of the weak law 4209: 4087: 3971: 3864: 3812: 3624:. Then for any fixed 3506: 3453: 3403: 3343: 3271: 3209: 3030: 2932: 2861: 2796: 2633: 2562: 2484: 2428: 2373:is likely to be near 2368: 2308: 2234: 2195: 2061: 1967: 1863: 1829: 1784: 1738:Gaussian distribution 1708: 1592: 1440:pairwise independence 1417: 1191: 1171: 1112:absolutely continuous 1071: 1001: 805:Bernoulli's principle 740: 637:approaches infinity. 582: 558:empirical probability 542: 393:that states that the 364: 171:Binomial distribution 10524:Probability theorems 10335:Probability theory 1 9897:, Chapter 1.4, p. 14 9154: 9090: 9030: 8970: 8891: 8836: 8749: 8689: 8571: 8333: 8281: 8232: 8212: 8063: 8043: 7982: 7956: 7936: 7845: 7787: 7749: 7704: 7684: 7577: 7550: 7490: 7445: 7336: 7213: 6999: 6960: 6894: 6823: 6736: 6710: 6568: 6473: 6381: 6302: 6274: 6214: 6156: 6104: 6073: 5973: 5892: 5853: 5636: 5591: 5429: 5317: 5202: 5018: 4911: 4875: 4817: 4589: 4567: 4515: 4415: 4321: 4243: 4141: 4118:and finite non-zero 4028: 4020:occurs in the first 3830: 3740: 3462: 3412: 3360: 3303: 3230: 3045: 2874: 2811: 2650: 2587: 2515: 2437: 2381: 2343: 2231: 2108: 1994: 1900: 1838: 1797: 1744: 1623: 1525: 1202: 1180: 1128: 1017: 928: 867:law of large numbers 692:Pareto distributions 650:, almost surely the 486: 383:law of large numbers 297:Law of large numbers 266:Marginal probability 191:Poisson distribution 40:Part of a series on 10529:Mathematical proofs 10342:Newey, Whitney K.; 10075:J. Geyer, Charles. 9573:The Drunkard's Walk 9110: 9081:= 1.028 when n=250 9050: 8990: 8853: 8781: 8420: 8159: 8009: 7872: 7817: 7731: 7653: 7619: 7604: 7532: 7517: 7472: 7404: 7363: 7258: 7230: 7026: 6241: 5193:, we have obtained 3511:), but for a given 2724: 2323:, Theorem 2.3.10). 2216:provided that each 1436:Mutual independence 902:Lebesgue integrable 801:Bernoulli's theorem 688:Cauchy distribution 652:absolute difference 256:Complementary event 198:Probability measure 186:Pareto distribution 181:Normal distribution 10472:Weisstein, Eric W. 10453:Weisstein, Eric W. 10410:10.3150/12-BEJSP12 10196:10.1007/BF01013465 10043:Mukherjee, Sayan. 9832:10.1007/BF01013465 9190: 9131: 9093: 9071: 9033: 9021:= 0.905 when n=25 9011: 8973: 8954: 8874: 8839: 8822: 8812: 8767: 8735: 8675: 8673: 8487:Monte Carlo Method 8461: 8394: 8317: 8244: 8218: 8198: 8145: 8049: 8029: 7995: 7968: 7942: 7922: 7858: 7831: 7803: 7783:terms of the form 7773: 7735: 7717: 7700:terms of the form 7690: 7667: 7639: 7605: 7590: 7563: 7536: 7518: 7503: 7476: 7458: 7431: 7390: 7349: 7322: 7244: 7216: 7199: 7150: 7012: 6982: 6946: 6880: 6819:Define the events 6809: 6796: 6722: 6696: 6691: 6586: 6554: 6505: 6459: 6413: 6367: 6329: 6286: 6257: 6227: 6200: 6142: 6086: 6035: 5948: 5873: 5829: 5611: 5566: 5487: 5385: 5264: 5173: 5000: 4895: 4855: 4799: 4643: 4573: 4553: 4477: 4394: 4354: 4305: 4204: 4082: 3980:Extremum estimator 3966: 3848: 3807: 3501: 3448: 3398: 3338: 3266: 3204: 3199: 3025: 2856: 2791: 2704: 2640:Dirichlet integral 2628: 2557: 2479: 2423: 2363: 2303: 2190: 2056: 2021: 1962: 1868:and goes to zero. 1858: 1824: 1779: 1703: 1641: 1587: 1412: 1256: 1186: 1166: 1066: 996: 777: 722:goes to infinity. 660:Monte Carlo method 586: 537: 476:. Therefore, the 379:probability theory 375: 307:Boole's inequality 243:Stochastic process 132:Mutual exclusivity 49:Probability theory 10366:978-0-13-603313-4 10259:978-3-540-74685-0 10138:Wen, Liu (1991). 9775:978-1-4939-4030-1 9442:978-1-85233-896-1 9410:10.1002/wics.1314 9188: 8952: 8811: 8672: 8362: 8221:{\displaystyle n} 8193: 8133: 8052:{\displaystyle n} 7945:{\displaystyle n} 7693:{\displaystyle n} 7111: 6795: 6690: 6685: 6671: 6613: 6571: 6532: 6490: 6440: 6398: 6339: 6314: 6059: 6058: 6022: 6017: 6008: 6004: 5995: 5985: 5933: 5923: 5904: 5865: 5814: 5769: 5749: 5703: 5653: 5603: 5584:are independent. 5496: 5486: 5445: 5301:complex functions 5288: 5287: 5251: 5246: 5237: 5233: 5224: 5214: 5168: 5104: 5044: 4995: 4937: 4887: 4835: 4794: 4774: 4701: 4642: 4610: 4576:{\displaystyle i} 4501: 4500: 4464: 4459: 4450: 4446: 4437: 4427: 4353: 4333: 4199: 4068: 4057: 3962: 3958: 3862: 3833: 3794: 3790: 3563:. But even after 3559:which depends on 3499: 3446: 3356:is asymptotic to 3264: 3226:be plus or minus 3178: 3107: 2989: 2931: 2920: 2789: 2757: 2703: 2692: 2532: 2454: 2398: 2355: 2270: 2214: 2213: 2183: 2179: 2178: 2170: 2153: 2120: 2031: 2006: 1986: 1985: 1949: 1944: 1935: 1931: 1930: 1922: 1912: 1666: 1626: 1611: 1610: 1574: 1569: 1560: 1556: 1547: 1537: 1407: 1387: 1314: 1255: 1223: 1189:{\displaystyle i} 1090: 1089: 1053: 1048: 1029: 959: 940: 529: 436:gambler's fallacy 359: 358: 261:Joint probability 208:Bernoulli process 107:Probability space 16:(Redirected from 10551: 10508:Business Insider 10485: 10484: 10466: 10465: 10447: 10421: 10403: 10394:(4): 1088–1121. 10379: 10370: 10351: 10344:McFadden, Daniel 10338: 10329: 10306: 10297: 10269: 10268: 10267: 10266: 10233: 10224: 10223: 10215: 10209: 10208: 10198: 10174: 10168: 10167: 10135: 10129: 10128: 10126: 10102: 10096: 10090: 10084: 10083: 10081: 10072: 10066: 10065: 10063: 10062: 10056: 10050:. Archived from 10049: 10040: 10034: 10033: 10031: 10030: 10024: 10009: 10000: 9994: 9993: 9973: 9967: 9962: 9956: 9955: 9953: 9952: 9938: 9932: 9931: 9922:Yuri Prokhorov. 9919: 9910: 9904: 9898: 9892: 9886: 9885: 9875: 9851: 9845: 9844: 9834: 9810: 9804: 9803: 9801: 9800: 9789: 9780: 9779: 9753: 9747: 9746: 9731: 9722: 9716: 9707: 9706: 9700: 9692: 9660: 9654: 9653: 9625: 9619: 9617: 9605: 9598: 9592: 9591: 9583: 9577: 9576: 9568: 9562: 9561: 9543: 9519: 9513: 9512: 9496: 9486: 9480: 9479: 9463: 9453: 9447: 9446: 9428: 9422: 9421: 9393: 9387: 9386: 9384: 9375: 9352: 9351: 9315: 9309: 9308: 9292: 9282: 9199: 9197: 9196: 9191: 9189: 9187: 9176: 9169: 9168: 9158: 9140: 9138: 9137: 9132: 9130: 9109: 9104: 9080: 9078: 9077: 9072: 9070: 9049: 9044: 9020: 9018: 9017: 9012: 9010: 8989: 8984: 8963: 8961: 8960: 8955: 8953: 8945: 8944: 8935: 8924: 8923: 8883: 8881: 8880: 8875: 8873: 8852: 8847: 8831: 8829: 8828: 8823: 8821: 8813: 8810: 8796: 8780: 8775: 8744: 8742: 8741: 8736: 8728: 8727: 8684: 8682: 8681: 8676: 8674: 8668: 8664: 8663: 8630: 8629: 8608: 8607: 8591: 8470: 8468: 8467: 8462: 8419: 8408: 8363: 8358: 8348: 8347: 8337: 8326: 8324: 8323: 8318: 8253: 8251: 8250: 8245: 8227: 8225: 8224: 8219: 8207: 8205: 8204: 8199: 8194: 8192: 8191: 8190: 8181: 8180: 8167: 8158: 8153: 8141: 8140: 8134: 8132: 8131: 8130: 8108: 8091: 8086: 8085: 8076: 8058: 8056: 8055: 8050: 8038: 8036: 8035: 8030: 8028: 8027: 8008: 8003: 7991: 7990: 7977: 7975: 7974: 7969: 7951: 7949: 7948: 7943: 7931: 7929: 7928: 7923: 7918: 7917: 7871: 7866: 7854: 7853: 7840: 7838: 7837: 7832: 7830: 7829: 7816: 7811: 7799: 7798: 7782: 7780: 7779: 7774: 7744: 7742: 7741: 7736: 7730: 7725: 7713: 7712: 7699: 7697: 7696: 7691: 7676: 7674: 7673: 7668: 7666: 7665: 7652: 7647: 7635: 7634: 7618: 7613: 7603: 7598: 7586: 7585: 7572: 7570: 7569: 7564: 7562: 7561: 7545: 7543: 7542: 7537: 7531: 7526: 7516: 7511: 7499: 7498: 7485: 7483: 7482: 7477: 7471: 7466: 7454: 7453: 7440: 7438: 7437: 7432: 7427: 7426: 7414: 7413: 7403: 7398: 7386: 7385: 7373: 7372: 7362: 7357: 7345: 7344: 7331: 7329: 7328: 7323: 7321: 7320: 7311: 7310: 7301: 7300: 7291: 7290: 7278: 7277: 7268: 7267: 7257: 7252: 7240: 7239: 7229: 7224: 7208: 7206: 7205: 7200: 7195: 7191: 7190: 7189: 7180: 7179: 7170: 7169: 7160: 7159: 7149: 7105: 7104: 7095: 7091: 7090: 7085: 7081: 7080: 7079: 7069: 7064: 7039: 7038: 7025: 7020: 7008: 7007: 6991: 6989: 6988: 6983: 6978: 6977: 6955: 6953: 6952: 6947: 6933: 6932: 6916: 6911: 6889: 6887: 6886: 6881: 6867: 6862: 6861: 6852: 6835: 6834: 6818: 6816: 6815: 6810: 6802: 6798: 6797: 6793: 6781: 6767: 6766: 6757: 6731: 6729: 6728: 6723: 6705: 6703: 6702: 6697: 6692: 6689:infinitely often 6688: 6683: 6676: 6672: 6667: 6657: 6656: 6646: 6614: 6609: 6599: 6598: 6588: 6585: 6563: 6561: 6560: 6555: 6544: 6540: 6533: 6528: 6518: 6517: 6507: 6504: 6468: 6466: 6465: 6460: 6452: 6448: 6441: 6436: 6426: 6425: 6415: 6412: 6376: 6374: 6373: 6368: 6357: 6353: 6346: 6345: 6340: 6332: 6328: 6295: 6293: 6292: 6287: 6266: 6264: 6263: 6258: 6240: 6235: 6223: 6222: 6209: 6207: 6206: 6201: 6193: 6192: 6177: 6176: 6151: 6149: 6148: 6143: 6126: 6125: 6113: 6112: 6095: 6093: 6092: 6087: 6085: 6084: 6053: 6044: 6042: 6041: 6036: 6020: 6019: 6018: 6015: 6006: 6005: 5997: 5993: 5992: 5991: 5986: 5978: 5967: 5957: 5955: 5954: 5949: 5934: 5931: 5924: 5922: 5914: 5911: 5910: 5905: 5897: 5882: 5880: 5879: 5874: 5872: 5871: 5866: 5858: 5838: 5836: 5835: 5830: 5815: 5812: 5806: 5805: 5785: 5784: 5779: 5775: 5774: 5770: 5762: 5750: 5742: 5719: 5718: 5713: 5709: 5708: 5704: 5696: 5690: 5689: 5662: 5661: 5660: 5659: 5654: 5646: 5620: 5618: 5617: 5612: 5610: 5609: 5604: 5596: 5575: 5573: 5572: 5567: 5555: 5554: 5536: 5535: 5514: 5513: 5497: 5494: 5488: 5479: 5473: 5472: 5451: 5450: 5446: 5438: 5394: 5392: 5391: 5386: 5329: 5328: 5297:Taylor's theorem 5282: 5273: 5271: 5270: 5265: 5249: 5248: 5247: 5244: 5235: 5234: 5226: 5222: 5221: 5220: 5215: 5207: 5196: 5182: 5180: 5179: 5174: 5169: 5167: 5166: 5165: 5152: 5151: 5142: 5122: 5118: 5111: 5110: 5105: 5097: 5062: 5058: 5051: 5050: 5045: 5037: 5009: 5007: 5006: 5001: 4996: 4994: 4993: 4992: 4979: 4978: 4969: 4955: 4951: 4944: 4943: 4938: 4930: 4904: 4902: 4901: 4896: 4894: 4893: 4888: 4880: 4864: 4862: 4861: 4856: 4842: 4841: 4836: 4828: 4808: 4806: 4805: 4800: 4795: 4790: 4789: 4780: 4775: 4773: 4772: 4763: 4762: 4761: 4748: 4740: 4739: 4721: 4720: 4702: 4700: 4699: 4687: 4676: 4675: 4657: 4656: 4644: 4635: 4617: 4616: 4611: 4603: 4582: 4580: 4579: 4574: 4562: 4560: 4559: 4554: 4552: 4551: 4536: 4535: 4495: 4486: 4484: 4483: 4478: 4462: 4461: 4460: 4457: 4448: 4447: 4439: 4435: 4434: 4433: 4428: 4420: 4409: 4403: 4401: 4400: 4395: 4387: 4386: 4368: 4367: 4355: 4346: 4340: 4339: 4334: 4326: 4314: 4312: 4311: 4306: 4283: 4282: 4261: 4260: 4213: 4211: 4210: 4205: 4200: 4198: 4197: 4185: 4168: 4154: 4134: 4091: 4089: 4088: 4083: 4069: 4066: 4058: 4053: 4043: 4042: 4032: 3975: 3973: 3972: 3967: 3960: 3959: 3957: 3949: 3947: 3943: 3900: 3899: 3883: 3878: 3863: 3855: 3847: 3816: 3814: 3813: 3808: 3792: 3791: 3788: 3770: 3766: 3628:, the sequence { 3555:)/2 before some 3510: 3508: 3507: 3502: 3500: 3474: 3472: 3457: 3455: 3454: 3449: 3447: 3424: 3416: 3407: 3405: 3404: 3399: 3376: 3347: 3345: 3344: 3339: 3313: 3289: 3288: 3284: 3275: 3273: 3272: 3267: 3265: 3242: 3234: 3213: 3211: 3210: 3205: 3203: 3202: 3179: 3177: 3145: 3108: 3106: 3080: 3034: 3032: 3031: 3026: 3003: 3002: 2990: 2985: 2984: 2983: 2965: 2964: 2954: 2951: 2946: 2929: 2925: 2921: 2916: 2915: 2914: 2896: 2895: 2885: 2865: 2863: 2862: 2857: 2855: 2854: 2842: 2841: 2823: 2822: 2800: 2798: 2797: 2792: 2790: 2782: 2771: 2770: 2758: 2753: 2752: 2751: 2726: 2723: 2718: 2701: 2697: 2693: 2688: 2687: 2686: 2661: 2644:Riemann integral 2637: 2635: 2634: 2629: 2627: 2626: 2614: 2613: 2566: 2564: 2563: 2558: 2550: 2539: 2538: 2533: 2525: 2522: 2510: 2500:shows that this 2488: 2486: 2485: 2480: 2472: 2461: 2460: 2455: 2447: 2444: 2432: 2430: 2429: 2424: 2416: 2405: 2404: 2399: 2391: 2388: 2372: 2370: 2369: 2364: 2362: 2361: 2356: 2348: 2312: 2310: 2309: 2304: 2290: 2289: 2271: 2269: 2268: 2256: 2253: 2248: 2208: 2199: 2197: 2196: 2191: 2181: 2180: 2176: 2172: 2168: 2167: 2166: 2160: 2159: 2154: 2146: 2143: 2142: 2127: 2126: 2121: 2113: 2102: 2065: 2063: 2062: 2057: 2049: 2045: 2038: 2037: 2032: 2024: 2020: 1980: 1971: 1969: 1968: 1963: 1947: 1946: 1945: 1942: 1933: 1932: 1928: 1924: 1920: 1919: 1918: 1913: 1905: 1894: 1867: 1865: 1864: 1859: 1848: 1833: 1831: 1830: 1825: 1814: 1809: 1808: 1788: 1786: 1785: 1780: 1757: 1712: 1710: 1709: 1704: 1696: 1692: 1684: 1673: 1672: 1667: 1659: 1656: 1640: 1605: 1596: 1594: 1593: 1588: 1572: 1571: 1570: 1567: 1558: 1557: 1549: 1545: 1544: 1543: 1538: 1530: 1519: 1487: 1478: 1469: 1421: 1419: 1418: 1413: 1408: 1403: 1402: 1393: 1388: 1386: 1385: 1376: 1375: 1374: 1361: 1353: 1352: 1334: 1333: 1315: 1313: 1312: 1300: 1289: 1288: 1270: 1269: 1257: 1248: 1230: 1229: 1224: 1216: 1195: 1193: 1192: 1187: 1175: 1173: 1172: 1167: 1165: 1164: 1149: 1148: 1116:Lebesgue measure 1114:with respect to 1084: 1075: 1073: 1072: 1067: 1051: 1050: 1049: 1046: 1036: 1035: 1030: 1022: 1011: 1005: 1003: 1002: 997: 992: 991: 973: 972: 960: 952: 947: 946: 941: 933: 881:of large numbers 874:of large numbers 820: 809:Daniel Bernoulli 794: 781:Gerolamo Cardano 768:phenomenon (see 649: 648: 644: 632: 631: 627: 611: 610: 606: 601: 600: 596: 562:Bernoulli trials 546: 544: 543: 538: 530: 525: 490: 391:mathematical law 351: 344: 337: 127:Elementary event 59: 37: 36: 21: 10559: 10558: 10554: 10553: 10552: 10550: 10549: 10548: 10514: 10513: 10432: 10429: 10424: 10367: 10326: 10294: 10277: 10272: 10264: 10262: 10260: 10234: 10227: 10216: 10212: 10175: 10171: 10156:10.2307/2323947 10136: 10132: 10103: 10099: 10091: 10087: 10079: 10073: 10069: 10060: 10058: 10054: 10047: 10041: 10037: 10028: 10026: 10022: 10007: 10001: 9997: 9990: 9974: 9970: 9963: 9959: 9950: 9948: 9940: 9939: 9935: 9920: 9913: 9905: 9901: 9893: 9889: 9852: 9848: 9811: 9807: 9798: 9796: 9791: 9790: 9783: 9776: 9754: 9750: 9732: 9725: 9717: 9710: 9694: 9693: 9675:(33): 259–267. 9661: 9657: 9642:10.2307/2709176 9626: 9622: 9599: 9595: 9584: 9580: 9569: 9565: 9520: 9516: 9509: 9487: 9483: 9476: 9454: 9450: 9443: 9429: 9425: 9394: 9390: 9382: 9376: 9355: 9316: 9312: 9305: 9283: 9272: 9268: 9230:Law of averages 9206: 9177: 9164: 9160: 9159: 9157: 9155: 9152: 9151: 9150: 9123: 9105: 9097: 9091: 9088: 9087: 9063: 9045: 9037: 9031: 9028: 9027: 9003: 8985: 8977: 8971: 8968: 8967: 8940: 8936: 8934: 8919: 8915: 8892: 8889: 8888: 8866: 8848: 8843: 8837: 8834: 8833: 8814: 8800: 8794: 8776: 8771: 8750: 8747: 8746: 8723: 8719: 8690: 8687: 8686: 8659: 8655: 8625: 8621: 8603: 8599: 8592: 8589: 8572: 8569: 8568: 8567:) by computing 8566: 8562: 8558: 8551: 8547: 8543: 8536: 8532: 8528: 8524: 8520: 8516: 8512: 8508: 8504: 8500: 8496: 8483: 8409: 8398: 8343: 8339: 8338: 8336: 8334: 8331: 8330: 8282: 8279: 8278: 8267: 8257: 8233: 8230: 8229: 8213: 8210: 8209: 8186: 8182: 8176: 8172: 8171: 8166: 8154: 8149: 8136: 8135: 8126: 8122: 8112: 8107: 8087: 8081: 8077: 8072: 8064: 8061: 8060: 8044: 8041: 8040: 8023: 8019: 8004: 7999: 7986: 7985: 7983: 7980: 7979: 7957: 7954: 7953: 7937: 7934: 7933: 7913: 7909: 7867: 7862: 7849: 7848: 7846: 7843: 7842: 7825: 7821: 7812: 7807: 7794: 7793: 7788: 7785: 7784: 7750: 7747: 7746: 7726: 7721: 7708: 7707: 7705: 7702: 7701: 7685: 7682: 7681: 7661: 7657: 7648: 7643: 7630: 7629: 7614: 7609: 7599: 7594: 7581: 7580: 7578: 7575: 7574: 7557: 7553: 7551: 7548: 7547: 7527: 7522: 7512: 7507: 7494: 7493: 7491: 7488: 7487: 7467: 7462: 7449: 7448: 7446: 7443: 7442: 7422: 7418: 7409: 7408: 7399: 7394: 7381: 7380: 7368: 7364: 7358: 7353: 7340: 7339: 7337: 7334: 7333: 7316: 7312: 7306: 7302: 7296: 7292: 7286: 7282: 7273: 7269: 7263: 7259: 7253: 7248: 7235: 7231: 7225: 7220: 7214: 7211: 7210: 7185: 7181: 7175: 7171: 7165: 7161: 7155: 7151: 7115: 7110: 7106: 7100: 7099: 7086: 7075: 7071: 7065: 7054: 7049: 7045: 7044: 7040: 7034: 7033: 7021: 7016: 7003: 7002: 7000: 6997: 6996: 6973: 6969: 6961: 6958: 6957: 6928: 6924: 6912: 6901: 6895: 6892: 6891: 6863: 6857: 6853: 6848: 6830: 6826: 6824: 6821: 6820: 6791: 6777: 6762: 6758: 6753: 6746: 6742: 6737: 6734: 6733: 6711: 6708: 6707: 6686: 6652: 6648: 6647: 6645: 6641: 6594: 6590: 6589: 6587: 6575: 6569: 6566: 6565: 6513: 6509: 6508: 6506: 6494: 6483: 6479: 6474: 6471: 6470: 6421: 6417: 6416: 6414: 6402: 6391: 6387: 6382: 6379: 6378: 6341: 6331: 6330: 6318: 6313: 6309: 6303: 6300: 6299: 6275: 6272: 6271: 6236: 6231: 6218: 6217: 6215: 6212: 6211: 6188: 6184: 6172: 6168: 6157: 6154: 6153: 6121: 6117: 6108: 6107: 6105: 6102: 6101: 6080: 6076: 6074: 6071: 6070: 6067: 6014: 6013: 5996: 5987: 5977: 5976: 5974: 5971: 5970: 5930: 5918: 5913: 5906: 5896: 5895: 5893: 5890: 5889: 5867: 5857: 5856: 5854: 5851: 5850: 5811: 5795: 5791: 5780: 5761: 5757: 5741: 5728: 5724: 5723: 5714: 5695: 5691: 5685: 5681: 5680: 5676: 5675: 5655: 5645: 5644: 5643: 5639: 5637: 5634: 5633: 5629: 5605: 5595: 5594: 5592: 5589: 5588: 5550: 5546: 5531: 5527: 5503: 5499: 5493: 5477: 5468: 5464: 5437: 5436: 5432: 5430: 5427: 5426: 5419: 5410: 5403: 5324: 5320: 5318: 5315: 5314: 5293: 5243: 5242: 5225: 5216: 5206: 5205: 5203: 5200: 5199: 5161: 5157: 5153: 5147: 5143: 5141: 5106: 5096: 5095: 5094: 5090: 5046: 5036: 5035: 5034: 5030: 5019: 5016: 5015: 4988: 4984: 4980: 4974: 4970: 4968: 4939: 4929: 4928: 4927: 4923: 4912: 4909: 4908: 4889: 4879: 4878: 4876: 4873: 4872: 4837: 4827: 4826: 4818: 4815: 4814: 4785: 4781: 4779: 4768: 4764: 4757: 4753: 4749: 4747: 4735: 4731: 4716: 4712: 4695: 4691: 4686: 4671: 4667: 4652: 4648: 4633: 4612: 4602: 4601: 4590: 4587: 4586: 4568: 4565: 4564: 4547: 4543: 4531: 4527: 4516: 4513: 4512: 4506: 4456: 4455: 4438: 4429: 4419: 4418: 4416: 4413: 4412: 4382: 4378: 4363: 4359: 4344: 4335: 4325: 4324: 4322: 4319: 4318: 4278: 4274: 4256: 4252: 4244: 4241: 4240: 4234: 4227: 4219: 4193: 4189: 4184: 4164: 4150: 4142: 4139: 4138: 4129: 4125:. Then for any 4109:random variable 4065: 4038: 4034: 4033: 4031: 4029: 4026: 4025: 4010: 3988: 3953: 3948: 3895: 3891: 3879: 3868: 3854: 3853: 3849: 3837: 3831: 3828: 3827: 3787: 3747: 3743: 3741: 3738: 3737: 3724:there exists a 3666:) convergence. 3653: 3638: 3591: 3586: 3473: 3468: 3463: 3460: 3459: 3420: 3415: 3413: 3410: 3409: 3372: 3361: 3358: 3357: 3309: 3304: 3301: 3300: 3298: 3286: 3282: 3281: 3238: 3233: 3231: 3228: 3227: 3225: 3198: 3197: 3183: 3149: 3144: 3139: 3124: 3123: 3112: 3084: 3079: 3074: 3049: 3048: 3046: 3043: 3042: 2995: 2991: 2979: 2975: 2960: 2956: 2955: 2953: 2947: 2936: 2910: 2906: 2891: 2887: 2886: 2884: 2880: 2875: 2872: 2871: 2847: 2843: 2837: 2833: 2818: 2814: 2812: 2809: 2808: 2781: 2763: 2759: 2747: 2743: 2727: 2725: 2719: 2708: 2682: 2678: 2662: 2660: 2656: 2651: 2648: 2647: 2619: 2615: 2609: 2605: 2588: 2585: 2584: 2546: 2534: 2524: 2523: 2518: 2516: 2513: 2512: 2511:the inequality 2505: 2468: 2456: 2446: 2445: 2440: 2438: 2435: 2434: 2412: 2400: 2390: 2389: 2384: 2382: 2379: 2378: 2357: 2347: 2346: 2344: 2341: 2340: 2329: 2285: 2281: 2264: 2260: 2255: 2249: 2238: 2232: 2229: 2228: 2224: 2171: 2162: 2161: 2155: 2145: 2144: 2138: 2137: 2122: 2112: 2111: 2109: 2106: 2105: 2033: 2023: 2022: 2010: 2005: 2001: 1995: 1992: 1991: 1941: 1940: 1923: 1914: 1904: 1903: 1901: 1898: 1897: 1877: 1844: 1839: 1836: 1835: 1810: 1804: 1800: 1798: 1795: 1794: 1753: 1745: 1742: 1741: 1680: 1668: 1658: 1657: 1652: 1650: 1646: 1630: 1624: 1621: 1620: 1566: 1565: 1548: 1539: 1529: 1528: 1526: 1523: 1522: 1498: 1497: 1496: 1495: 1490: 1489: 1488: 1480: 1479: 1471: 1470: 1459: 1444:exchangeability 1398: 1394: 1392: 1381: 1377: 1370: 1366: 1362: 1360: 1348: 1344: 1329: 1325: 1308: 1304: 1299: 1284: 1280: 1265: 1261: 1246: 1225: 1215: 1214: 1203: 1200: 1199: 1181: 1178: 1177: 1160: 1156: 1144: 1140: 1129: 1126: 1125: 1104: 1097: 1045: 1044: 1031: 1021: 1020: 1018: 1015: 1014: 987: 983: 968: 964: 951: 942: 932: 931: 929: 926: 925: 917: 910: 896: 889: 863: 791:Ars Conjectandi 785:Jacob Bernoulli 775: 735: 680: 671:random sampling 646: 642: 641: 629: 625: 624: 608: 604: 603: 598: 594: 593: 588:For example, a 491: 489: 487: 484: 483: 470: 355: 203:Random variable 154:Bernoulli trial 35: 32: 23: 22: 15: 12: 11: 5: 10557: 10547: 10546: 10541: 10536: 10531: 10526: 10512: 10511: 10499: 10486: 10467: 10448: 10428: 10427:External links 10425: 10423: 10422: 10384:Seneta, Eugene 10380: 10371: 10365: 10352: 10346:(1994). "36". 10339: 10330: 10324: 10307: 10298: 10292: 10278: 10276: 10273: 10271: 10270: 10258: 10225: 10210: 10169: 10150:(2): 146–148. 10130: 10117:(2): 633–643. 10097: 10085: 10067: 10035: 9995: 9988: 9968: 9957: 9933: 9911: 9899: 9887: 9846: 9825:(1): 119–122. 9805: 9781: 9774: 9748: 9723: 9708: 9655: 9636:(3): 455–475. 9620: 9593: 9578: 9563: 9534:(3): 916–918. 9514: 9507: 9481: 9474: 9448: 9441: 9423: 9404:(6): 386–392. 9388: 9378:Sedor, Kelly. 9353: 9326:(3): 615–621. 9310: 9303: 9269: 9267: 9264: 9263: 9262: 9257: 9252: 9247: 9242: 9237: 9232: 9227: 9222: 9217: 9212: 9205: 9202: 9186: 9183: 9180: 9175: 9172: 9167: 9163: 9148: 9129: 9126: 9122: 9119: 9116: 9113: 9108: 9103: 9100: 9096: 9069: 9066: 9062: 9059: 9056: 9053: 9048: 9043: 9040: 9036: 9009: 9006: 9002: 8999: 8996: 8993: 8988: 8983: 8980: 8976: 8951: 8948: 8943: 8939: 8933: 8930: 8927: 8922: 8918: 8914: 8911: 8908: 8905: 8902: 8899: 8896: 8885: 8884: 8872: 8869: 8865: 8862: 8859: 8856: 8851: 8846: 8842: 8820: 8817: 8809: 8806: 8803: 8799: 8793: 8790: 8787: 8784: 8779: 8774: 8770: 8766: 8763: 8760: 8757: 8754: 8734: 8731: 8726: 8722: 8718: 8715: 8712: 8709: 8706: 8703: 8700: 8697: 8694: 8671: 8667: 8662: 8658: 8654: 8651: 8648: 8645: 8642: 8639: 8636: 8633: 8628: 8624: 8620: 8617: 8614: 8611: 8606: 8602: 8598: 8595: 8588: 8585: 8582: 8579: 8576: 8564: 8560: 8556: 8553: 8549: 8545: 8541: 8538: 8534: 8530: 8526: 8522: 8518: 8514: 8510: 8506: 8502: 8498: 8494: 8482: 8479: 8460: 8457: 8454: 8451: 8448: 8445: 8442: 8439: 8436: 8432: 8429: 8426: 8423: 8418: 8415: 8412: 8407: 8404: 8401: 8397: 8393: 8390: 8387: 8384: 8381: 8378: 8375: 8372: 8369: 8366: 8361: 8357: 8354: 8351: 8346: 8342: 8316: 8313: 8310: 8307: 8304: 8301: 8298: 8295: 8292: 8289: 8286: 8273:. By applying 8266: 8263: 8243: 8240: 8237: 8217: 8197: 8189: 8185: 8179: 8175: 8170: 8165: 8162: 8157: 8152: 8148: 8144: 8139: 8129: 8125: 8121: 8118: 8115: 8111: 8106: 8103: 8100: 8097: 8094: 8090: 8084: 8080: 8075: 8071: 8068: 8048: 8026: 8022: 8018: 8015: 8012: 8007: 8002: 7998: 7994: 7989: 7967: 7964: 7961: 7941: 7921: 7916: 7912: 7908: 7905: 7902: 7899: 7896: 7893: 7890: 7887: 7884: 7881: 7878: 7875: 7870: 7865: 7861: 7857: 7852: 7828: 7824: 7820: 7815: 7810: 7806: 7802: 7797: 7792: 7772: 7769: 7766: 7763: 7760: 7757: 7754: 7734: 7729: 7724: 7720: 7716: 7711: 7689: 7664: 7660: 7656: 7651: 7646: 7642: 7638: 7633: 7628: 7625: 7622: 7617: 7612: 7608: 7602: 7597: 7593: 7589: 7584: 7560: 7556: 7535: 7530: 7525: 7521: 7515: 7510: 7506: 7502: 7497: 7475: 7470: 7465: 7461: 7457: 7452: 7430: 7425: 7421: 7417: 7412: 7407: 7402: 7397: 7393: 7389: 7384: 7379: 7376: 7371: 7367: 7361: 7356: 7352: 7348: 7343: 7319: 7315: 7309: 7305: 7299: 7295: 7289: 7285: 7281: 7276: 7272: 7266: 7262: 7256: 7251: 7247: 7243: 7238: 7234: 7228: 7223: 7219: 7198: 7194: 7188: 7184: 7178: 7174: 7168: 7164: 7158: 7154: 7148: 7145: 7142: 7139: 7136: 7133: 7130: 7127: 7124: 7121: 7118: 7114: 7109: 7103: 7098: 7094: 7089: 7084: 7078: 7074: 7068: 7063: 7060: 7057: 7053: 7048: 7043: 7037: 7032: 7029: 7024: 7019: 7015: 7011: 7006: 6981: 6976: 6972: 6968: 6965: 6945: 6942: 6939: 6936: 6931: 6927: 6923: 6920: 6915: 6910: 6907: 6904: 6900: 6879: 6876: 6873: 6870: 6866: 6860: 6856: 6851: 6847: 6844: 6841: 6838: 6833: 6829: 6808: 6805: 6801: 6790: 6787: 6784: 6780: 6776: 6773: 6770: 6765: 6761: 6756: 6752: 6749: 6745: 6741: 6721: 6718: 6715: 6695: 6682: 6679: 6675: 6670: 6666: 6663: 6660: 6655: 6651: 6644: 6640: 6637: 6634: 6631: 6628: 6624: 6620: 6617: 6612: 6608: 6605: 6602: 6597: 6593: 6584: 6581: 6578: 6574: 6553: 6550: 6547: 6543: 6539: 6536: 6531: 6527: 6524: 6521: 6516: 6512: 6503: 6500: 6497: 6493: 6489: 6486: 6482: 6478: 6458: 6455: 6451: 6447: 6444: 6439: 6435: 6432: 6429: 6424: 6420: 6411: 6408: 6405: 6401: 6397: 6394: 6390: 6386: 6366: 6363: 6360: 6356: 6352: 6349: 6344: 6338: 6335: 6327: 6324: 6321: 6317: 6312: 6307: 6285: 6282: 6279: 6256: 6253: 6250: 6247: 6244: 6239: 6234: 6230: 6226: 6221: 6199: 6196: 6191: 6187: 6183: 6180: 6175: 6171: 6167: 6164: 6161: 6141: 6138: 6135: 6132: 6129: 6124: 6120: 6116: 6111: 6083: 6079: 6066: 6063: 6057: 6056: 6047: 6045: 6034: 6031: 6028: 6025: 6011: 6003: 6000: 5990: 5984: 5981: 5964:.) Therefore, 5947: 5944: 5941: 5938: 5928: 5921: 5917: 5909: 5903: 5900: 5870: 5864: 5861: 5828: 5825: 5822: 5819: 5809: 5804: 5801: 5798: 5794: 5789: 5783: 5778: 5773: 5768: 5765: 5760: 5756: 5753: 5748: 5745: 5740: 5737: 5734: 5731: 5727: 5722: 5717: 5712: 5707: 5702: 5699: 5694: 5688: 5684: 5679: 5674: 5671: 5668: 5665: 5658: 5652: 5649: 5642: 5625: 5608: 5602: 5599: 5564: 5561: 5558: 5553: 5549: 5545: 5542: 5539: 5534: 5530: 5526: 5523: 5520: 5517: 5512: 5509: 5506: 5502: 5491: 5485: 5482: 5476: 5471: 5467: 5463: 5460: 5457: 5454: 5449: 5444: 5441: 5435: 5415: 5408: 5401: 5384: 5381: 5378: 5374: 5371: 5368: 5365: 5362: 5359: 5356: 5353: 5350: 5347: 5344: 5341: 5338: 5335: 5332: 5327: 5323: 5292: 5289: 5286: 5285: 5276: 5274: 5263: 5260: 5257: 5254: 5240: 5232: 5229: 5219: 5213: 5210: 5172: 5164: 5160: 5156: 5150: 5146: 5140: 5137: 5134: 5131: 5128: 5125: 5121: 5117: 5114: 5109: 5103: 5100: 5093: 5089: 5086: 5083: 5080: 5077: 5074: 5071: 5068: 5065: 5061: 5057: 5054: 5049: 5043: 5040: 5033: 5029: 5026: 5023: 4999: 4991: 4987: 4983: 4977: 4973: 4967: 4964: 4961: 4958: 4954: 4950: 4947: 4942: 4936: 4933: 4926: 4922: 4919: 4916: 4892: 4886: 4883: 4854: 4851: 4848: 4845: 4840: 4834: 4831: 4825: 4822: 4798: 4793: 4788: 4784: 4778: 4771: 4767: 4760: 4756: 4752: 4746: 4743: 4738: 4734: 4730: 4727: 4724: 4719: 4715: 4711: 4708: 4705: 4698: 4694: 4690: 4685: 4682: 4679: 4674: 4670: 4666: 4663: 4660: 4655: 4651: 4647: 4641: 4638: 4632: 4629: 4626: 4623: 4620: 4615: 4609: 4606: 4600: 4597: 4594: 4572: 4550: 4546: 4542: 4539: 4534: 4530: 4526: 4523: 4520: 4505: 4502: 4499: 4498: 4489: 4487: 4476: 4473: 4470: 4467: 4453: 4445: 4442: 4432: 4426: 4423: 4393: 4390: 4385: 4381: 4377: 4374: 4371: 4366: 4362: 4358: 4352: 4349: 4343: 4338: 4332: 4329: 4304: 4301: 4298: 4295: 4292: 4289: 4286: 4281: 4277: 4273: 4270: 4267: 4264: 4259: 4255: 4251: 4248: 4232: 4225: 4218: 4215: 4203: 4196: 4192: 4188: 4183: 4180: 4177: 4174: 4171: 4167: 4163: 4160: 4157: 4153: 4149: 4146: 4113:expected value 4081: 4078: 4075: 4072: 4067: as 4064: 4061: 4056: 4052: 4049: 4046: 4041: 4037: 4008: 3993:, named after 3987: 3984: 3965: 3956: 3952: 3946: 3942: 3939: 3936: 3933: 3930: 3927: 3924: 3921: 3918: 3915: 3912: 3909: 3906: 3903: 3898: 3894: 3890: 3887: 3882: 3877: 3874: 3871: 3867: 3861: 3858: 3852: 3846: 3843: 3840: 3836: 3818: 3817: 3806: 3803: 3800: 3797: 3785: 3782: 3779: 3776: 3773: 3769: 3765: 3762: 3759: 3756: 3753: 3750: 3746: 3722: 3689: 3651: 3636: 3590: 3587: 3585: 3584: 3498: 3495: 3492: 3489: 3486: 3483: 3480: 3477: 3471: 3467: 3445: 3442: 3439: 3436: 3433: 3430: 3427: 3423: 3419: 3397: 3394: 3391: 3388: 3385: 3382: 3379: 3375: 3371: 3368: 3365: 3337: 3334: 3331: 3328: 3325: 3322: 3319: 3316: 3312: 3308: 3294: 3263: 3260: 3257: 3254: 3251: 3248: 3245: 3241: 3237: 3221: 3215: 3201: 3196: 3193: 3190: 3187: 3184: 3182: 3176: 3173: 3170: 3167: 3164: 3161: 3158: 3155: 3152: 3148: 3143: 3140: 3138: 3135: 3132: 3129: 3126: 3125: 3122: 3119: 3116: 3113: 3111: 3105: 3102: 3099: 3096: 3093: 3090: 3087: 3083: 3078: 3075: 3073: 3070: 3067: 3064: 3061: 3058: 3055: 3054: 3052: 3035: 3024: 3021: 3018: 3015: 3012: 3009: 3006: 3001: 2998: 2994: 2988: 2982: 2978: 2974: 2971: 2968: 2963: 2959: 2950: 2945: 2942: 2939: 2935: 2928: 2924: 2919: 2913: 2909: 2905: 2902: 2899: 2894: 2890: 2883: 2879: 2853: 2850: 2846: 2840: 2836: 2832: 2829: 2826: 2821: 2817: 2801: 2788: 2785: 2780: 2777: 2774: 2769: 2766: 2762: 2756: 2750: 2746: 2742: 2739: 2736: 2733: 2730: 2722: 2717: 2714: 2711: 2707: 2700: 2696: 2691: 2685: 2681: 2677: 2674: 2671: 2668: 2665: 2659: 2655: 2646:, we can say: 2625: 2622: 2618: 2612: 2608: 2604: 2601: 2598: 2595: 2592: 2576: 2556: 2553: 2549: 2545: 2542: 2537: 2531: 2528: 2521: 2478: 2475: 2471: 2467: 2464: 2459: 2453: 2450: 2443: 2422: 2419: 2415: 2411: 2408: 2403: 2397: 2394: 2387: 2360: 2354: 2351: 2339:, the average 2328: 2325: 2302: 2299: 2296: 2293: 2288: 2284: 2280: 2277: 2274: 2267: 2263: 2259: 2252: 2247: 2244: 2241: 2237: 2220: 2212: 2211: 2202: 2200: 2189: 2186: 2175: 2165: 2158: 2152: 2149: 2141: 2136: 2133: 2130: 2125: 2119: 2116: 2055: 2052: 2048: 2044: 2041: 2036: 2030: 2027: 2019: 2016: 2013: 2009: 2004: 1999: 1984: 1983: 1974: 1972: 1961: 1958: 1955: 1952: 1938: 1927: 1917: 1911: 1908: 1876: 1873: 1857: 1854: 1851: 1847: 1843: 1823: 1820: 1817: 1813: 1807: 1803: 1778: 1775: 1772: 1769: 1766: 1763: 1760: 1756: 1752: 1749: 1702: 1699: 1695: 1690: 1687: 1683: 1679: 1676: 1671: 1665: 1662: 1655: 1649: 1644: 1639: 1636: 1633: 1629: 1609: 1608: 1599: 1597: 1586: 1583: 1580: 1577: 1563: 1555: 1552: 1542: 1536: 1533: 1492: 1491: 1482: 1481: 1473: 1472: 1464: 1463: 1462: 1461: 1460: 1458: 1455: 1411: 1406: 1401: 1397: 1391: 1384: 1380: 1373: 1369: 1365: 1359: 1356: 1351: 1347: 1343: 1340: 1337: 1332: 1328: 1324: 1321: 1318: 1311: 1307: 1303: 1298: 1295: 1292: 1287: 1283: 1279: 1276: 1273: 1268: 1264: 1260: 1254: 1251: 1245: 1242: 1239: 1236: 1233: 1228: 1222: 1219: 1213: 1210: 1207: 1185: 1163: 1159: 1155: 1152: 1147: 1143: 1139: 1136: 1133: 1102: 1095: 1088: 1087: 1078: 1076: 1065: 1062: 1059: 1056: 1042: 1039: 1034: 1028: 1025: 995: 990: 986: 982: 979: 976: 971: 967: 963: 958: 955: 950: 945: 939: 936: 915: 908: 894: 887: 862: 859: 850:expected value 774: 773: 762: 756: 749: 734: 731: 727:selection bias 679: 676: 536: 533: 528: 524: 521: 518: 515: 512: 509: 506: 503: 500: 497: 494: 478:expected value 469: 466: 357: 356: 354: 353: 346: 339: 331: 328: 327: 326: 325: 320: 312: 311: 310: 309: 304: 302:Bayes' theorem 299: 294: 289: 284: 276: 275: 274: 273: 268: 263: 258: 250: 249: 248: 247: 246: 245: 240: 235: 233:Observed value 230: 225: 220: 218:Expected value 215: 210: 200: 195: 194: 193: 188: 183: 178: 173: 168: 158: 157: 156: 146: 145: 144: 139: 134: 129: 124: 114: 109: 101: 100: 99: 98: 93: 88: 87: 86: 76: 75: 74: 61: 60: 52: 51: 45: 44: 33: 9: 6: 4: 3: 2: 10556: 10545: 10544:Large numbers 10542: 10540: 10537: 10535: 10532: 10530: 10527: 10525: 10522: 10521: 10519: 10510: 10509: 10503: 10500: 10498: 10494: 10490: 10487: 10482: 10481: 10476: 10473: 10468: 10463: 10462: 10457: 10454: 10449: 10445: 10441: 10440: 10435: 10431: 10430: 10419: 10415: 10411: 10407: 10402: 10397: 10393: 10389: 10385: 10381: 10377: 10372: 10368: 10362: 10358: 10353: 10349: 10345: 10340: 10336: 10331: 10327: 10325:87-91180-71-6 10321: 10317: 10313: 10308: 10304: 10299: 10295: 10293:0-19-853665-8 10289: 10285: 10280: 10279: 10261: 10255: 10251: 10247: 10243: 10239: 10232: 10230: 10221: 10214: 10206: 10202: 10197: 10192: 10188: 10184: 10180: 10173: 10165: 10161: 10157: 10153: 10149: 10145: 10141: 10134: 10125: 10120: 10116: 10112: 10108: 10101: 10094: 10089: 10078: 10071: 10057:on 2013-03-09 10053: 10046: 10039: 10025:on 2016-07-01 10021: 10017: 10013: 10006: 9999: 9991: 9989:9780387276052 9985: 9981: 9980: 9972: 9966: 9961: 9947: 9943: 9937: 9929: 9925: 9918: 9916: 9908: 9903: 9896: 9891: 9883: 9879: 9874: 9869: 9865: 9861: 9857: 9850: 9842: 9838: 9833: 9828: 9824: 9820: 9816: 9809: 9794: 9788: 9786: 9777: 9771: 9767: 9763: 9759: 9752: 9744: 9740: 9736: 9735:Yuri Prohorov 9730: 9728: 9720: 9715: 9713: 9704: 9698: 9690: 9686: 9682: 9678: 9674: 9671:(in French). 9670: 9666: 9659: 9651: 9647: 9643: 9639: 9635: 9631: 9624: 9615: 9611: 9604: 9597: 9589: 9582: 9574: 9567: 9559: 9555: 9551: 9547: 9542: 9537: 9533: 9529: 9525: 9518: 9510: 9508:9781852338961 9504: 9500: 9495: 9494: 9485: 9477: 9475:9781852338961 9471: 9467: 9462: 9461: 9452: 9444: 9438: 9434: 9427: 9419: 9415: 9411: 9407: 9403: 9399: 9392: 9381: 9374: 9372: 9370: 9368: 9366: 9364: 9362: 9360: 9358: 9349: 9345: 9341: 9337: 9333: 9329: 9325: 9321: 9314: 9306: 9304:9781852338961 9300: 9296: 9291: 9290: 9281: 9279: 9277: 9275: 9270: 9261: 9258: 9256: 9253: 9251: 9248: 9246: 9243: 9241: 9238: 9236: 9233: 9231: 9228: 9226: 9223: 9221: 9218: 9216: 9213: 9211: 9208: 9207: 9201: 9184: 9181: 9178: 9173: 9170: 9165: 9161: 9145: 9142: 9127: 9124: 9117: 9111: 9106: 9101: 9098: 9094: 9085: 9082: 9067: 9064: 9057: 9051: 9046: 9041: 9038: 9034: 9025: 9022: 9007: 9004: 8997: 8991: 8986: 8981: 8978: 8974: 8965: 8949: 8946: 8941: 8937: 8928: 8920: 8916: 8912: 8909: 8906: 8900: 8894: 8870: 8867: 8860: 8854: 8849: 8844: 8840: 8818: 8815: 8807: 8804: 8801: 8797: 8788: 8782: 8777: 8772: 8768: 8761: 8758: 8755: 8724: 8720: 8713: 8707: 8701: 8698: 8695: 8669: 8660: 8656: 8649: 8646: 8643: 8640: 8637: 8634: 8626: 8622: 8615: 8612: 8604: 8600: 8593: 8583: 8580: 8577: 8554: 8539: 8492: 8491: 8490: 8488: 8478: 8476: 8471: 8455: 8449: 8446: 8443: 8437: 8434: 8427: 8421: 8416: 8413: 8410: 8405: 8402: 8399: 8395: 8391: 8385: 8382: 8379: 8373: 8370: 8367: 8359: 8352: 8344: 8340: 8328: 8311: 8308: 8305: 8302: 8299: 8296: 8293: 8287: 8284: 8276: 8272: 8262: 8259: 8255: 8241: 8238: 8235: 8215: 8195: 8187: 8183: 8177: 8173: 8168: 8163: 8155: 8150: 8146: 8127: 8119: 8116: 8109: 8104: 8098: 8095: 8092: 8082: 8078: 8046: 8024: 8020: 8016: 8013: 8005: 8000: 7996: 7965: 7962: 7959: 7939: 7919: 7914: 7910: 7903: 7900: 7897: 7891: 7888: 7885: 7882: 7879: 7876: 7868: 7863: 7859: 7826: 7813: 7808: 7804: 7767: 7764: 7761: 7755: 7752: 7727: 7722: 7718: 7687: 7678: 7662: 7649: 7644: 7640: 7623: 7615: 7610: 7606: 7600: 7595: 7591: 7558: 7554: 7546:. Since the 7528: 7523: 7519: 7513: 7508: 7504: 7468: 7463: 7459: 7423: 7419: 7400: 7395: 7391: 7377: 7369: 7365: 7359: 7354: 7350: 7317: 7313: 7307: 7303: 7297: 7293: 7287: 7283: 7279: 7274: 7270: 7264: 7260: 7254: 7249: 7245: 7241: 7236: 7232: 7226: 7221: 7217: 7196: 7192: 7186: 7182: 7176: 7172: 7166: 7162: 7156: 7152: 7146: 7143: 7140: 7137: 7134: 7131: 7128: 7125: 7122: 7119: 7116: 7112: 7107: 7096: 7092: 7087: 7082: 7076: 7072: 7066: 7061: 7058: 7055: 7051: 7046: 7041: 7030: 7022: 7017: 7013: 6993: 6974: 6970: 6943: 6937: 6929: 6925: 6908: 6905: 6902: 6898: 6874: 6871: 6868: 6858: 6854: 6845: 6842: 6836: 6831: 6827: 6806: 6803: 6799: 6788: 6785: 6782: 6771: 6763: 6759: 6750: 6747: 6743: 6719: 6716: 6713: 6693: 6680: 6677: 6673: 6668: 6661: 6653: 6649: 6642: 6638: 6635: 6632: 6629: 6618: 6615: 6610: 6603: 6595: 6591: 6576: 6551: 6548: 6545: 6541: 6537: 6534: 6529: 6522: 6514: 6510: 6495: 6487: 6484: 6480: 6456: 6453: 6449: 6445: 6442: 6437: 6430: 6422: 6418: 6403: 6395: 6392: 6388: 6364: 6361: 6358: 6354: 6350: 6347: 6342: 6333: 6319: 6310: 6297: 6283: 6280: 6277: 6268: 6251: 6248: 6245: 6237: 6232: 6228: 6194: 6189: 6185: 6181: 6173: 6169: 6162: 6159: 6136: 6133: 6130: 6122: 6118: 6099: 6081: 6077: 6062: 6055: 6048: 6046: 6032: 6023: 6009: 6001: 5988: 5979: 5969: 5968: 5965: 5963: 5958: 5945: 5936: 5926: 5907: 5898: 5887: 5885: 5868: 5859: 5848: 5844: 5839: 5826: 5817: 5807: 5802: 5799: 5796: 5792: 5781: 5776: 5771: 5766: 5763: 5758: 5754: 5751: 5746: 5743: 5738: 5735: 5732: 5729: 5725: 5720: 5715: 5710: 5705: 5700: 5697: 5692: 5686: 5682: 5677: 5672: 5666: 5656: 5647: 5640: 5631: 5628: 5624: 5606: 5597: 5585: 5583: 5579: 5559: 5551: 5547: 5540: 5532: 5528: 5524: 5518: 5510: 5507: 5504: 5500: 5483: 5480: 5469: 5465: 5461: 5455: 5447: 5442: 5439: 5433: 5424: 5421: 5418: 5414: 5407: 5400: 5395: 5382: 5376: 5372: 5366: 5360: 5357: 5354: 5351: 5348: 5345: 5342: 5339: 5333: 5325: 5321: 5312: 5310: 5306: 5302: 5298: 5284: 5277: 5275: 5261: 5252: 5238: 5230: 5217: 5208: 5198: 5197: 5194: 5192: 5188: 5183: 5170: 5162: 5158: 5154: 5148: 5144: 5138: 5135: 5132: 5126: 5123: 5119: 5115: 5112: 5107: 5098: 5091: 5084: 5078: 5075: 5072: 5066: 5063: 5059: 5055: 5052: 5047: 5038: 5031: 5024: 5013: 5010: 4997: 4989: 4985: 4981: 4975: 4971: 4965: 4959: 4956: 4952: 4948: 4945: 4940: 4931: 4924: 4917: 4906: 4890: 4881: 4870: 4865: 4852: 4849: 4846: 4838: 4829: 4820: 4812: 4809: 4796: 4791: 4786: 4782: 4776: 4769: 4765: 4758: 4754: 4750: 4744: 4736: 4732: 4728: 4725: 4722: 4717: 4713: 4706: 4703: 4696: 4692: 4688: 4683: 4672: 4668: 4664: 4661: 4658: 4653: 4649: 4639: 4636: 4627: 4624: 4621: 4613: 4604: 4595: 4592: 4584: 4570: 4548: 4544: 4540: 4532: 4528: 4521: 4518: 4511: 4497: 4490: 4488: 4474: 4465: 4451: 4443: 4430: 4421: 4411: 4410: 4407: 4404: 4391: 4383: 4379: 4375: 4372: 4369: 4364: 4360: 4350: 4347: 4341: 4336: 4327: 4316: 4299: 4296: 4293: 4290: 4287: 4279: 4275: 4268: 4265: 4257: 4253: 4246: 4238: 4231: 4224: 4214: 4201: 4194: 4190: 4186: 4181: 4175: 4172: 4169: 4161: 4158: 4155: 4136: 4132: 4128: 4124: 4121: 4117: 4114: 4110: 4106: 4102: 4101: 4096: 4092: 4079: 4070: 4062: 4054: 4047: 4039: 4035: 4023: 4019: 4015: 4011: 4004: 4000: 3996: 3992: 3983: 3981: 3976: 3963: 3934: 3931: 3928: 3922: 3916: 3910: 3904: 3901: 3896: 3892: 3885: 3880: 3875: 3872: 3869: 3865: 3859: 3856: 3841: 3838: 3825: 3823: 3804: 3798: 3795: 3780: 3774: 3771: 3760: 3757: 3754: 3748: 3735: 3731: 3727: 3723: 3720: 3716: 3712: 3709: 3705: 3701: 3697: 3693: 3690: 3687: 3684: 3683: 3682: 3680: 3676: 3672: 3667: 3665: 3661: 3657: 3650: 3646: 3642: 3635: 3631: 3627: 3623: 3619: 3615: 3611: 3607: 3603: 3598: 3596: 3582: 3578: 3574: 3570: 3566: 3562: 3558: 3554: 3550: 3546: 3542: 3538: 3534: 3530: 3526: 3522: 3518: 3514: 3496: 3493: 3490: 3487: 3484: 3481: 3478: 3475: 3469: 3465: 3443: 3440: 3437: 3434: 3431: 3428: 3425: 3421: 3417: 3395: 3392: 3389: 3386: 3383: 3380: 3377: 3373: 3369: 3366: 3363: 3355: 3352: = 3351: 3335: 3332: 3329: 3326: 3323: 3320: 3317: 3314: 3310: 3306: 3297: 3293: 3279: 3261: 3258: 3255: 3252: 3249: 3246: 3243: 3239: 3235: 3224: 3220: 3216: 3194: 3191: 3188: 3185: 3180: 3171: 3168: 3162: 3159: 3156: 3153: 3150: 3146: 3141: 3133: 3127: 3120: 3117: 3114: 3109: 3100: 3094: 3091: 3088: 3085: 3081: 3076: 3068: 3062: 3059: 3056: 3050: 3040: 3036: 3019: 3013: 3010: 3007: 3004: 2999: 2996: 2992: 2986: 2980: 2972: 2969: 2961: 2957: 2943: 2940: 2937: 2933: 2926: 2922: 2917: 2911: 2903: 2900: 2892: 2888: 2881: 2877: 2869: 2851: 2848: 2844: 2838: 2830: 2827: 2819: 2815: 2806: 2805:geometrically 2802: 2786: 2783: 2778: 2775: 2772: 2767: 2764: 2760: 2754: 2748: 2744: 2737: 2731: 2728: 2715: 2712: 2709: 2705: 2698: 2694: 2689: 2683: 2679: 2672: 2666: 2663: 2657: 2653: 2645: 2641: 2623: 2620: 2616: 2610: 2606: 2599: 2593: 2590: 2582: 2581:exponentially 2578: 2577: 2575: 2572: 2570: 2554: 2551: 2543: 2540: 2535: 2526: 2508: 2503: 2502:almost surely 2499: 2494: 2492: 2476: 2473: 2465: 2462: 2457: 2448: 2420: 2417: 2409: 2406: 2401: 2392: 2376: 2358: 2349: 2338: 2334: 2324: 2322: 2318: 2313: 2300: 2294: 2286: 2282: 2275: 2272: 2265: 2261: 2257: 2245: 2242: 2239: 2235: 2226: 2223: 2219: 2210: 2203: 2201: 2187: 2184: 2156: 2147: 2134: 2128: 2123: 2114: 2104: 2103: 2100: 2097: 2095: 2089: 2087: 2081: 2079: 2074: 2071: 2066: 2053: 2050: 2046: 2042: 2039: 2034: 2025: 2011: 2002: 1989: 1982: 1975: 1973: 1959: 1950: 1936: 1915: 1906: 1896: 1895: 1892: 1890: 1886: 1883:(also called 1882: 1872: 1869: 1855: 1852: 1849: 1845: 1841: 1821: 1818: 1815: 1811: 1805: 1801: 1792: 1773: 1770: 1767: 1761: 1758: 1754: 1750: 1747: 1739: 1735: 1731: 1728:of the first 1726: 1720: 1718: 1713: 1700: 1697: 1693: 1688: 1685: 1677: 1674: 1669: 1660: 1647: 1631: 1618: 1616: 1607: 1600: 1598: 1584: 1575: 1561: 1553: 1540: 1531: 1521: 1520: 1517: 1515: 1511: 1507: 1504:(also called 1503: 1486: 1477: 1468: 1454: 1452: 1447: 1445: 1441: 1437: 1433: 1431: 1430:not necessary 1427: 1422: 1409: 1404: 1399: 1395: 1389: 1382: 1378: 1371: 1367: 1363: 1357: 1349: 1345: 1341: 1338: 1335: 1330: 1326: 1319: 1316: 1309: 1305: 1301: 1296: 1285: 1281: 1277: 1274: 1271: 1266: 1262: 1252: 1249: 1240: 1237: 1234: 1226: 1217: 1208: 1205: 1197: 1183: 1161: 1157: 1153: 1145: 1141: 1134: 1131: 1124: 1119: 1117: 1113: 1109: 1105: 1098: 1086: 1079: 1077: 1063: 1054: 1040: 1032: 1023: 1013: 1012: 1009: 1006: 988: 984: 980: 977: 974: 969: 965: 956: 953: 948: 943: 934: 923: 921: 914: 907: 903: 900: 893: 886: 882: 880: 875: 873: 868: 858: 856: 851: 847: 843: 839: 835: 831: 827: 822: 819: 814: 813:S. D. Poisson 810: 806: 802: 798: 793: 792: 786: 782: 771: 766: 763: 760: 757: 754: 751: 750: 747: 743: 739: 730: 728: 723: 721: 717: 713: 709: 705: 701: 697: 693: 689: 685: 675: 672: 668: 665: 664:computational 661: 656: 653: 638: 636: 622: 619: 618:almost surely 615: 591: 581: 577: 575: 571: 567: 563: 559: 554: 552: 547: 534: 531: 526: 522: 519: 516: 513: 510: 507: 504: 501: 498: 495: 492: 481: 479: 475: 465: 462: 460: 456: 452: 448: 444: 439: 437: 432: 427: 423: 419: 415: 412: 407: 405: 401: 396: 392: 388: 384: 380: 372: 368: 363: 352: 347: 345: 340: 338: 333: 332: 330: 329: 324: 321: 319: 316: 315: 314: 313: 308: 305: 303: 300: 298: 295: 293: 290: 288: 285: 283: 280: 279: 278: 277: 272: 269: 267: 264: 262: 259: 257: 254: 253: 252: 251: 244: 241: 239: 236: 234: 231: 229: 226: 224: 221: 219: 216: 214: 211: 209: 206: 205: 204: 201: 199: 196: 192: 189: 187: 184: 182: 179: 177: 174: 172: 169: 167: 164: 163: 162: 159: 155: 152: 151: 150: 147: 143: 140: 138: 135: 133: 130: 128: 125: 123: 120: 119: 118: 115: 113: 110: 108: 105: 104: 103: 102: 97: 94: 92: 91:Indeterminism 89: 85: 82: 81: 80: 77: 73: 70: 69: 68: 65: 64: 63: 62: 58: 54: 53: 50: 47: 46: 43: 39: 38: 30: 19: 10505: 10478: 10459: 10437: 10391: 10387: 10375: 10356: 10347: 10334: 10315: 10311: 10302: 10283: 10263:, retrieved 10241: 10219: 10213: 10186: 10182: 10172: 10147: 10143: 10133: 10114: 10110: 10100: 10088: 10070: 10059:. Retrieved 10052:the original 10038: 10027:. Retrieved 10020:the original 10015: 10011: 9998: 9982:. Springer. 9978: 9971: 9960: 9949:. Retrieved 9945: 9936: 9927: 9902: 9890: 9863: 9859: 9849: 9822: 9818: 9808: 9797:. Retrieved 9757: 9751: 9745:. EMS Press. 9742: 9697:cite journal 9672: 9668: 9658: 9633: 9629: 9623: 9609: 9596: 9587: 9581: 9572: 9566: 9531: 9527: 9517: 9492: 9484: 9459: 9451: 9432: 9426: 9401: 9397: 9391: 9323: 9319: 9313: 9288: 9245:Lindy effect 9146: 9143: 9086: 9083: 9026: 9023: 8966: 8886: 8540:Evaluate f(X 8521:= a+(b - a)U 8484: 8481:Applications 8472: 8329: 8268: 8265:Consequences 8260: 8256: 7679: 6995:We compute 6994: 6298: 6269: 6068: 6060: 6049: 5959: 5888: 5842: 5840: 5632: 5626: 5622: 5621:in terms of 5586: 5581: 5577: 5425: 5422: 5416: 5412: 5405: 5398: 5396: 5313: 5308: 5294: 5278: 5186: 5184: 5014: 5011: 4907: 4866: 4813: 4810: 4585: 4507: 4491: 4405: 4317: 4229: 4222: 4220: 4137: 4130: 4122: 4115: 4111:with finite 4104: 4098: 4097: 4093: 4021: 4017: 4013: 4006: 4002: 3998: 3990: 3989: 3977: 3826: 3821: 3819: 3733: 3729: 3718: 3714: 3710: 3703: 3699: 3695: 3691: 3685: 3678: 3674: 3670: 3668: 3663: 3659: 3655: 3648: 3644: 3640: 3633: 3629: 3625: 3621: 3617: 3616:defined for 3609: 3605: 3601: 3599: 3594: 3592: 3580: 3576: 3572: 3568: 3564: 3560: 3556: 3552: 3548: 3544: 3540: 3536: 3532: 3528: 3524: 3520: 3516: 3512: 3353: 3349: 3295: 3291: 3277: 3222: 3218: 2579:Let X be an 2573: 2568: 2506: 2497: 2495: 2490: 2374: 2336: 2332: 2330: 2316: 2314: 2227: 2221: 2217: 2215: 2204: 2098: 2093: 2090: 2082: 2075: 2069: 2067: 1990: 1987: 1976: 1880: 1878: 1870: 1733: 1729: 1721: 1716: 1714: 1619: 1614: 1612: 1601: 1501: 1499: 1448: 1434: 1429: 1423: 1198: 1120: 1107: 1100: 1093: 1091: 1080: 1007: 924: 919: 912: 905: 891: 884: 878: 877: 871: 870: 866: 864: 823: 800: 796: 778: 764: 758: 752: 724: 719: 715: 703: 695: 683: 681: 657: 639: 634: 613: 587: 569: 555: 548: 482: 471: 463: 458: 454: 450: 446: 442: 440: 431:large number 430: 408: 386: 382: 376: 367:illustration 323:Tree diagram 318:Venn diagram 296: 282:Independence 228:Markov chain 112:Sample space 10095:, Lemma 2.4 9965:Ross (2009) 9946:builtin.com 9719:Seneta 2013 9141:= 1.000194 8563:), ..., f(X 8548:), ..., f(X 4905:results in 4127:real number 3995:Émile Borel 3688:is compact, 2803:Let X be a 2319:, see e.g. 855:convergence 811:. In 1837, 770:Fick's laws 700:heavy tails 616:flips will 551:sample mean 474:probability 461:increases. 400:sample mean 238:Random walk 79:Determinism 67:Probability 10518:Categories 10506:explained 10275:References 10265:2023-12-08 10061:2014-06-28 10029:2014-06-28 9951:2023-10-20 9907:Loève 1977 9895:Loève 1977 9799:2012-06-09 7978:such that 7680:There are 6732:, we have 6564:Note that 5841:The limit 3726:dominating 3708:almost all 3612:) is some 2498:strong law 1885:Kolmogorov 1875:Strong law 1791:asymptotic 918:) = ... = 872:strong law 842:Kolmogorov 678:Limitation 667:algorithms 149:Experiment 96:Randomness 42:statistics 10497:animation 10480:MathWorld 10461:MathWorld 10444:EMS Press 10401:1309.6488 10388:Bernoulli 10205:122166046 9882:0091-1798 9841:122166046 9689:120850863 9550:0003-4851 9340:1063-6706 9255:Sortition 9182:− 9171:− 9099:− 9095:∫ 9039:− 9035:∫ 8979:− 8975:∫ 8841:∫ 8805:− 8769:∫ 8759:− 8699:− 8581:− 8475:histogram 8441:≈ 8403:− 8396:∫ 8383:∈ 8365:≈ 8297:− 8236:ϵ 8174:ϵ 8164:≤ 8120:ϵ 8105:≤ 8099:ϵ 8093:≥ 8014:≤ 7911:σ 7901:− 7883:τ 7841:, and so 7765:− 7144:≤ 7120:≤ 7113:∑ 7052:∑ 6941:∞ 6914:∞ 6899:∑ 6875:ϵ 6869:≥ 6843:ω 6789:ϵ 6783:≥ 6772:ω 6748:ω 6714:ϵ 6681:ϵ 6678:≥ 6662:ω 6630:ϵ 6627:∃ 6623:⟺ 6616:≠ 6604:ω 6583:∞ 6580:→ 6535:≠ 6523:ω 6502:∞ 6499:→ 6485:ω 6431:ω 6410:∞ 6407:→ 6393:ω 6337:¯ 6326:∞ 6323:→ 6278:μ 6255:∞ 6249:τ 6198:∞ 6186:σ 6163:⁡ 6140:∞ 6134:μ 6030:∞ 6027:→ 6010:μ 5999:→ 5983:¯ 5943:∞ 5940:→ 5927:μ 5916:→ 5902:¯ 5863:¯ 5824:∞ 5821:→ 5803:μ 5788:→ 5739:μ 5683:φ 5651:¯ 5641:φ 5601:¯ 5548:φ 5529:φ 5501:φ 5466:φ 5434:φ 5380:→ 5355:μ 5322:φ 5259:∞ 5256:→ 5239:μ 5228:→ 5212:¯ 5159:ε 5145:σ 5139:− 5133:≥ 5127:ε 5124:≥ 5116:μ 5113:− 5102:¯ 5085:⁡ 5079:− 5067:ε 5056:μ 5053:− 5042:¯ 5025:⁡ 4986:ε 4972:σ 4966:≤ 4960:ε 4957:≥ 4949:μ 4946:− 4935:¯ 4918:⁡ 4885:¯ 4850:μ 4833:¯ 4783:σ 4755:σ 4726:⋯ 4707:⁡ 4662:⋯ 4628:⁡ 4608:¯ 4596:⁡ 4563:(for all 4545:σ 4522:⁡ 4472:∞ 4469:→ 4452:μ 4441:→ 4425:¯ 4373:⋯ 4331:¯ 4303:∞ 4297:μ 4291:⋯ 4182:≤ 4176:σ 4170:≥ 4162:μ 4159:− 4077:∞ 4074:→ 4060:→ 3951:→ 3935:θ 3917:⁡ 3911:− 3905:θ 3866:∑ 3845:Θ 3842:∈ 3839:θ 3802:Θ 3799:∈ 3796:θ 3772:≤ 3761:θ 3728:function 3675:uniformly 3660:pointwise 3494:⁡ 3488:⁡ 3482:⁡ 3441:⁡ 3435:⁡ 3429:⁡ 3393:⁡ 3387:⁡ 3381:⁡ 3367:⁡ 3330:⁡ 3324:⁡ 3318:⁡ 3259:⁡ 3253:⁡ 3247:⁡ 3192:− 3189:≤ 3169:− 3163:⁡ 3151:− 3118:≥ 3095:⁡ 3060:− 3014:⁡ 3008:− 2997:− 2970:− 2949:∞ 2934:∑ 2901:− 2849:− 2828:− 2784:π 2765:− 2732:⁡ 2721:∞ 2706:∫ 2667:⁡ 2621:− 2594:⁡ 2555:ε 2544:μ 2541:− 2530:¯ 2474:≠ 2466:μ 2463:− 2452:¯ 2421:ε 2410:μ 2407:− 2396:¯ 2353:¯ 2298:∞ 2276:⁡ 2251:∞ 2236:∑ 2174:⟶ 2151:¯ 2135:⁡ 2129:− 2118:¯ 2094:something 2043:μ 2029:¯ 2018:∞ 2015:→ 1988:That is, 1957:∞ 1954:→ 1937:μ 1926:⟶ 1910:¯ 1853:⁡ 1819:⁡ 1762:⁡ 1725:Chebyshev 1723:shown by 1689:ε 1678:μ 1675:− 1664:¯ 1638:∞ 1635:→ 1582:∞ 1579:→ 1562:μ 1551:→ 1535:¯ 1396:σ 1368:σ 1339:⋯ 1320:⁡ 1275:⋯ 1241:⁡ 1221:¯ 1209:⁡ 1176:(for all 1158:σ 1135:⁡ 1061:∞ 1058:→ 1041:μ 1038:→ 1027:¯ 978:⋯ 938:¯ 826:Chebyshev 742:Diffusion 590:fair coin 420:may lose 142:Singleton 10495:package 10418:88520834 9418:18521840 9204:See also 8533:, ..., X 8513:, ..., U 8501:, ..., X 4510:variance 4120:variance 3945:‖ 3851:‖ 3768:‖ 3745:‖ 3717:at each 3706:∈ Θ for 3614:function 3600:Suppose 3299:is then 2489:for all 2333:weak law 1506:Khinchin 1457:Weak law 1426:variance 1123:variance 879:weak law 876:and the 846:Khinchin 838:Cantelli 690:or some 621:converge 468:Examples 426:roulette 223:Variance 10446:, 2001 10164:2323947 9650:2709176 9558:2239008 9348:2238905 3789:for all 3285:⁄ 3037:If the 765:Bottom: 759:Middle: 733:History 708:tangent 645:⁄ 628:⁄ 607:⁄ 597:⁄ 443:average 395:average 389:) is a 137:Outcome 10416: 10363: 10322: 10290: 10256: 10203: 10162: 9986: 9880: 9839: 9772: 9687: 9648: 9606:) in: 9556: 9548: 9505: 9472: 9439: 9416: 9346: 9338: 9301: 9297:–190. 9149:f(x) = 8559:), f(X 8544:), f(X 6684: 6210:, and 6021: 6007: 5994: 5886:to μ: 5303:, the 5250: 5236: 5223: 4867:Using 4463: 4449: 4436: 4237:i.i.d. 4221:Given 4133:> 0 4103:. Let 3961: 3824:, and 3793: 2930: 2868:series 2702: 2509:> 0 2182: 2169: 1948: 1934: 1921: 1573: 1559: 1546: 1052: 911:) = E( 830:Markov 746:solute 712:median 418:casino 414:events 411:random 381:, the 84:System 72:Axioms 10414:S2CID 10396:arXiv 10314:[ 10201:S2CID 10160:JSTOR 10080:(PDF) 10055:(PDF) 10048:(PDF) 10023:(PDF) 10008:(PDF) 9866:(2). 9837:S2CID 9685:S2CID 9646:JSTOR 9554:JSTOR 9414:S2CID 9383:(PDF) 9344:S2CID 9266:Notes 4107:be a 3681:. If 861:Forms 834:Borel 422:money 117:Event 10361:ISBN 10320:ISBN 10288:ISBN 10254:ISBN 9984:ISBN 9878:ISSN 9770:ISBN 9703:link 9673:1846 9546:ISSN 9503:ISBN 9470:ISBN 9437:ISBN 9336:ISSN 9299:ISBN 9024:and 8239:> 8208:for 8039:for 7963:> 7745:and 7677:. 7486:and 6938:< 6717:> 6633:> 6252:< 6195:< 6137:< 6096:are 6016:when 5580:and 5397:All 5299:for 5245:when 5064:< 4458:when 4300:< 3662:(in 3217:Let 2552:< 2496:The 2418:> 2331:The 2295:< 2177:a.s. 2084:See 1943:when 1929:a.s. 1879:The 1686:< 1568:when 1500:The 844:and 753:Top: 404:mean 10406:doi 10246:doi 10191:doi 10152:doi 10119:doi 9868:doi 9827:doi 9762:doi 9677:doi 9638:doi 9536:doi 9406:doi 9328:doi 9295:181 8529:, X 8509:, U 8497:, X 6573:lim 6492:lim 6400:lim 6377:or 6316:lim 6160:Var 6098:iid 5932:for 5576:if 5495:and 5295:By 5185:As 4871:on 4704:Var 4625:Var 4593:Var 4519:Var 3982:). 3835:sup 3677:in 3643:), 3491:log 3485:log 3479:log 3438:log 3432:log 3426:log 3390:log 3384:log 3378:log 3364:log 3327:log 3321:log 3315:log 3256:log 3250:log 3244:log 2729:sin 2664:sin 2591:sin 2493:). 2273:Var 2008:lim 1850:log 1816:log 1793:to 1759:log 1628:lim 1442:or 1428:is 1317:Var 1238:Var 1206:Var 1132:Var 1118:.) 1108:not 633:as 623:to 535:3.5 457:as 449:of 447:sum 438:). 406:. 387:LLN 377:In 371:die 365:An 10520:: 10477:. 10458:. 10442:, 10436:, 10412:. 10404:. 10392:19 10390:. 10252:, 10240:, 10228:^ 10199:. 10187:55 10185:. 10181:. 10158:. 10148:98 10146:. 10142:. 10115:40 10113:. 10109:. 10016:13 10014:. 10010:. 9944:. 9926:. 9914:^ 9876:. 9862:. 9858:. 9835:. 9823:55 9821:. 9817:. 9784:^ 9768:. 9741:. 9737:. 9726:^ 9711:^ 9699:}} 9695:{{ 9667:. 9644:. 9634:44 9632:. 9552:. 9544:. 9532:38 9530:. 9526:. 9501:. 9499:63 9468:. 9466:92 9412:. 9400:. 9356:^ 9342:. 9334:. 9324:24 9322:. 9273:^ 8745:= 8477:. 8067:Pr 6992:. 6964:Pr 6919:Pr 6807:0. 6740:Pr 6477:Pr 6457:1. 6385:Pr 6306:Pr 6267:. 6246:=: 6152:, 6131:=: 6100:, 5849:, 5813:as 5630:: 5420:. 5404:, 5383:0. 4228:, 4145:Pr 4135:, 3964:0. 3597:. 3160:ln 3092:ln 3011:ln 2088:. 2054:1. 1998:Pr 1701:1. 1643:Pr 1617:, 1453:. 1047:as 890:, 840:, 836:, 832:, 828:, 10493:R 10483:. 10464:. 10420:. 10408:: 10398:: 10369:. 10328:. 10296:. 10248:: 10222:. 10207:. 10193:: 10166:. 10154:: 10127:. 10121:: 10082:. 10064:. 10032:. 9992:. 9954:. 9930:. 9884:. 9870:: 9864:6 9843:. 9829:: 9802:. 9778:. 9764:: 9721:. 9705:) 9691:. 9679:: 9652:. 9640:: 9616:. 9614:7 9560:. 9538:: 9511:. 9478:. 9445:. 9420:. 9408:: 9402:6 9385:. 9350:. 9330:: 9307:. 9185:1 9179:e 9174:1 9166:x 9162:e 9128:x 9125:d 9121:) 9118:x 9115:( 9112:f 9107:2 9102:1 9068:x 9065:d 9061:) 9058:x 9055:( 9052:f 9047:2 9042:1 9008:x 9005:d 9001:) 8998:x 8995:( 8992:f 8987:2 8982:1 8950:1 8947:+ 8942:3 8938:x 8932:) 8929:x 8926:( 8921:2 8917:s 8913:o 8910:c 8907:= 8904:) 8901:x 8898:( 8895:f 8871:x 8868:d 8864:) 8861:x 8858:( 8855:f 8850:b 8845:a 8832:= 8819:x 8816:d 8808:a 8802:b 8798:1 8792:) 8789:x 8786:( 8783:f 8778:b 8773:a 8765:) 8762:a 8756:b 8753:( 8733:) 8730:) 8725:1 8721:X 8717:( 8714:f 8711:( 8708:E 8705:) 8702:a 8696:b 8693:( 8670:n 8666:) 8661:n 8657:X 8653:( 8650:f 8647:+ 8644:. 8641:. 8638:. 8635:+ 8632:) 8627:2 8623:X 8619:( 8616:f 8613:+ 8610:) 8605:1 8601:X 8597:( 8594:f 8587:) 8584:a 8578:b 8575:( 8565:n 8561:2 8557:1 8552:) 8550:n 8546:2 8542:1 8535:n 8531:2 8527:1 8523:i 8519:i 8515:n 8511:2 8507:1 8503:n 8499:2 8495:1 8459:) 8456:a 8453:( 8450:f 8447:h 8444:2 8438:x 8435:d 8431:) 8428:x 8425:( 8422:f 8417:h 8414:+ 8411:a 8406:h 8400:a 8392:= 8389:) 8386:C 8380:X 8377:( 8374:P 8371:= 8368:p 8360:n 8356:) 8353:C 8350:( 8345:n 8341:N 8315:] 8312:h 8309:+ 8306:a 8303:, 8300:h 8294:a 8291:( 8288:= 8285:C 8242:0 8216:n 8196:, 8188:2 8184:n 8178:4 8169:C 8161:] 8156:4 8151:n 8147:S 8143:[ 8138:E 8128:4 8124:) 8117:n 8114:( 8110:1 8102:) 8096:n 8089:| 8083:n 8079:S 8074:| 8070:( 8047:n 8025:2 8021:n 8017:C 8011:] 8006:4 8001:n 7997:S 7993:[ 7988:E 7966:0 7960:C 7940:n 7920:. 7915:4 7907:) 7904:1 7898:n 7895:( 7892:n 7889:3 7886:+ 7880:n 7877:= 7874:] 7869:4 7864:n 7860:S 7856:[ 7851:E 7827:2 7823:) 7819:] 7814:2 7809:i 7805:X 7801:[ 7796:E 7791:( 7771:) 7768:1 7762:n 7759:( 7756:n 7753:3 7733:] 7728:4 7723:i 7719:X 7715:[ 7710:E 7688:n 7663:2 7659:) 7655:] 7650:2 7645:i 7641:X 7637:[ 7632:E 7627:( 7624:= 7621:] 7616:2 7611:j 7607:X 7601:2 7596:i 7592:X 7588:[ 7583:E 7559:i 7555:X 7534:] 7529:2 7524:j 7520:X 7514:2 7509:i 7505:X 7501:[ 7496:E 7474:] 7469:4 7464:i 7460:X 7456:[ 7451:E 7429:] 7424:j 7420:X 7416:[ 7411:E 7406:] 7401:3 7396:i 7392:X 7388:[ 7383:E 7378:= 7375:] 7370:j 7366:X 7360:3 7355:i 7351:X 7347:[ 7342:E 7318:l 7314:X 7308:k 7304:X 7298:j 7294:X 7288:i 7284:X 7280:, 7275:k 7271:X 7265:j 7261:X 7255:2 7250:i 7246:X 7242:, 7237:j 7233:X 7227:3 7222:i 7218:X 7197:. 7193:] 7187:l 7183:X 7177:k 7173:X 7167:j 7163:X 7157:i 7153:X 7147:n 7141:l 7138:, 7135:k 7132:, 7129:j 7126:, 7123:i 7117:1 7108:[ 7102:E 7097:= 7093:] 7088:4 7083:) 7077:i 7073:X 7067:n 7062:1 7059:= 7056:i 7047:( 7042:[ 7036:E 7031:= 7028:] 7023:4 7018:n 7014:S 7010:[ 7005:E 6980:) 6975:n 6971:A 6967:( 6944:, 6935:) 6930:n 6926:A 6922:( 6909:1 6906:= 6903:n 6878:} 6872:n 6865:| 6859:n 6855:S 6850:| 6846:: 6840:{ 6837:= 6832:n 6828:A 6804:= 6800:) 6786:n 6779:| 6775:) 6769:( 6764:n 6760:S 6755:| 6751:: 6744:( 6720:0 6694:, 6674:| 6669:n 6665:) 6659:( 6654:n 6650:S 6643:| 6639:, 6636:0 6619:0 6611:n 6607:) 6601:( 6596:n 6592:S 6577:n 6552:, 6549:0 6546:= 6542:) 6538:0 6530:n 6526:) 6520:( 6515:n 6511:S 6496:n 6488:: 6481:( 6454:= 6450:) 6446:0 6443:= 6438:n 6434:) 6428:( 6423:n 6419:S 6404:n 6396:: 6389:( 6365:, 6362:1 6359:= 6355:) 6351:0 6348:= 6343:n 6334:X 6320:n 6311:( 6284:0 6281:= 6243:] 6238:4 6233:i 6229:X 6225:[ 6220:E 6190:2 6182:= 6179:) 6174:i 6170:X 6166:( 6128:] 6123:i 6119:X 6115:[ 6110:E 6082:i 6078:X 6054:) 6052:2 6050:( 6033:. 6024:n 6002:P 5989:n 5980:X 5946:. 5937:n 5920:D 5908:n 5899:X 5869:n 5860:X 5843:e 5827:. 5818:n 5808:, 5800:t 5797:i 5793:e 5782:n 5777:] 5772:) 5767:n 5764:t 5759:( 5755:o 5752:+ 5747:n 5744:t 5736:i 5733:+ 5730:1 5726:[ 5721:= 5716:n 5711:] 5706:) 5701:n 5698:t 5693:( 5687:X 5678:[ 5673:= 5670:) 5667:t 5664:( 5657:n 5648:X 5627:X 5623:φ 5607:n 5598:X 5582:Y 5578:X 5563:) 5560:t 5557:( 5552:Y 5544:) 5541:t 5538:( 5533:X 5525:= 5522:) 5519:t 5516:( 5511:Y 5508:+ 5505:X 5490:) 5484:n 5481:t 5475:( 5470:X 5462:= 5459:) 5456:t 5453:( 5448:X 5443:n 5440:1 5417:X 5413:φ 5409:2 5406:X 5402:1 5399:X 5377:t 5373:, 5370:) 5367:t 5364:( 5361:o 5358:+ 5352:t 5349:i 5346:+ 5343:1 5340:= 5337:) 5334:t 5331:( 5326:X 5309:X 5283:) 5281:2 5279:( 5262:. 5253:n 5231:P 5218:n 5209:X 5187:n 5171:. 5163:2 5155:n 5149:2 5136:1 5130:) 5120:| 5108:n 5099:X 5092:| 5088:( 5082:P 5076:1 5073:= 5070:) 5060:| 5048:n 5039:X 5032:| 5028:( 5022:P 4998:. 4990:2 4982:n 4976:2 4963:) 4953:| 4941:n 4932:X 4925:| 4921:( 4915:P 4891:n 4882:X 4853:. 4847:= 4844:) 4839:n 4830:X 4824:( 4821:E 4797:. 4792:n 4787:2 4777:= 4770:2 4766:n 4759:2 4751:n 4745:= 4742:) 4737:n 4733:X 4729:+ 4723:+ 4718:1 4714:X 4710:( 4697:2 4693:n 4689:1 4684:= 4681:) 4678:) 4673:n 4669:X 4665:+ 4659:+ 4654:1 4650:X 4646:( 4640:n 4637:1 4631:( 4622:= 4619:) 4614:n 4605:X 4599:( 4571:i 4549:2 4541:= 4538:) 4533:i 4529:X 4525:( 4496:) 4494:2 4492:( 4475:. 4466:n 4444:P 4431:n 4422:X 4392:. 4389:) 4384:n 4380:X 4376:+ 4370:+ 4365:1 4361:X 4357:( 4351:n 4348:1 4342:= 4337:n 4328:X 4294:= 4288:= 4285:) 4280:2 4276:X 4272:( 4269:E 4266:= 4263:) 4258:1 4254:X 4250:( 4247:E 4233:2 4230:X 4226:1 4223:X 4202:. 4195:2 4191:k 4187:1 4179:) 4173:k 4166:| 4156:X 4152:| 4148:( 4131:k 4123:σ 4116:μ 4105:X 4080:. 4071:n 4063:p 4055:n 4051:) 4048:E 4045:( 4040:n 4036:N 4022:n 4018:E 4014:E 4012:( 4009:n 4007:N 4003:p 3999:E 3955:P 3941:] 3938:) 3932:, 3929:X 3926:( 3923:f 3920:[ 3914:E 3908:) 3902:, 3897:i 3893:X 3889:( 3886:f 3881:n 3876:1 3873:= 3870:i 3860:n 3857:1 3822:θ 3805:. 3784:) 3781:x 3778:( 3775:d 3764:) 3758:, 3755:x 3752:( 3749:f 3734:x 3732:( 3730:d 3721:. 3719:θ 3715:x 3711:x 3704:θ 3700:θ 3698:, 3696:x 3694:( 3692:f 3686:Θ 3679:θ 3664:θ 3656:θ 3654:, 3652:2 3649:X 3647:( 3645:f 3641:θ 3639:, 3637:1 3634:X 3632:( 3630:f 3626:θ 3622:θ 3618:θ 3610:θ 3608:, 3606:x 3604:( 3602:f 3581:ε 3579:( 3577:p 3573:ε 3571:( 3569:p 3565:m 3561:n 3557:m 3553:ε 3551:( 3549:p 3545:n 3541:ε 3539:( 3537:p 3533:ε 3531:( 3529:p 3525:ε 3521:n 3517:n 3513:ε 3497:n 3476:2 3470:/ 3466:1 3444:k 3422:/ 3418:k 3396:n 3374:/ 3370:n 3354:n 3350:k 3336:. 3333:k 3311:/ 3307:k 3296:k 3292:X 3287:2 3283:1 3278:k 3262:k 3240:/ 3236:k 3223:k 3219:X 3195:e 3186:x 3181:, 3175:) 3172:x 3166:( 3157:x 3154:2 3147:e 3142:= 3137:) 3134:x 3131:( 3128:F 3121:e 3115:x 3110:, 3104:) 3101:x 3098:( 3089:x 3086:2 3082:e 3077:= 3072:) 3069:x 3066:( 3063:F 3057:1 3051:{ 3023:) 3020:2 3017:( 3005:= 3000:x 2993:2 2987:x 2981:x 2977:) 2973:1 2967:( 2962:x 2958:2 2944:1 2941:= 2938:x 2927:= 2923:) 2918:X 2912:X 2908:) 2904:1 2898:( 2893:X 2889:2 2882:( 2878:E 2852:1 2845:X 2839:X 2835:) 2831:1 2825:( 2820:X 2816:2 2787:2 2779:= 2776:x 2773:d 2768:x 2761:e 2755:x 2749:x 2745:e 2741:) 2738:x 2735:( 2716:0 2713:= 2710:x 2699:= 2695:) 2690:X 2684:X 2680:e 2676:) 2673:X 2670:( 2658:( 2654:E 2624:1 2617:X 2611:X 2607:e 2603:) 2600:X 2597:( 2569:n 2548:| 2536:n 2527:X 2520:| 2507:ε 2491:n 2477:0 2470:| 2458:n 2449:X 2442:| 2414:| 2402:n 2393:X 2386:| 2375:μ 2359:n 2350:X 2337:n 2301:. 2292:] 2287:k 2283:X 2279:[ 2266:2 2262:k 2258:1 2246:1 2243:= 2240:k 2222:k 2218:X 2209:) 2207:2 2205:( 2188:, 2185:0 2164:] 2157:n 2148:X 2140:[ 2132:E 2124:n 2115:X 2070:n 2051:= 2047:) 2040:= 2035:n 2026:X 2012:n 2003:( 1981:) 1979:3 1977:( 1960:. 1951:n 1916:n 1907:X 1856:n 1846:/ 1842:1 1822:n 1812:/ 1806:2 1802:n 1777:) 1774:1 1771:+ 1768:n 1765:( 1755:/ 1751:n 1748:2 1734:n 1730:n 1717:ε 1698:= 1694:) 1682:| 1670:n 1661:X 1654:| 1648:( 1632:n 1615:ε 1606:) 1604:2 1602:( 1585:. 1576:n 1554:P 1541:n 1532:X 1410:. 1405:n 1400:2 1390:= 1383:2 1379:n 1372:2 1364:n 1358:= 1355:) 1350:n 1346:X 1342:+ 1336:+ 1331:1 1327:X 1323:( 1310:2 1306:n 1302:1 1297:= 1294:) 1291:) 1286:n 1282:X 1278:+ 1272:+ 1267:1 1263:X 1259:( 1253:n 1250:1 1244:( 1235:= 1232:) 1227:n 1218:X 1212:( 1184:i 1162:2 1154:= 1151:) 1146:i 1142:X 1138:( 1103:j 1101:X 1096:j 1094:X 1085:) 1083:1 1081:( 1064:. 1055:n 1033:n 1024:X 994:) 989:n 985:X 981:+ 975:+ 970:1 966:X 962:( 957:n 954:1 949:= 944:n 935:X 920:μ 916:2 913:X 909:1 906:X 895:2 892:X 888:1 885:X 795:( 720:n 716:n 704:α 696:n 684:n 647:2 643:1 635:n 630:2 626:1 614:n 609:2 605:1 599:2 595:1 570:n 532:= 527:6 523:6 520:+ 517:5 514:+ 511:4 508:+ 505:3 502:+ 499:2 496:+ 493:1 459:n 455:n 451:n 385:( 350:e 343:t 336:v 31:. 20:)

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Knowledge

Law of large numbers

Index