Law of large numbers

569: 46: 10493:. "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. 351: 1474: 1456: 4796: 1409: 3986:, 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 5170: 8266:, 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: 6693: 4577: 1190: 3963: 727: 5006: 1716:

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

5563: 362:. 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. 3201: 4997: 5826: 2788: 8195: 837:. 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 1959: 6556: 5945: 3022: 6032: 5261: 4474: 2187: 1584: 3818: 8458: 772:(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 417:

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.

4791:{\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}}.} 4083:

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.

1404:{\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}}.} 662:

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

1063: 5624: 4391: 2638: 8051: 7319: 7196: 5165:{\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}}}.} 993: 1778:, 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 422:

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

4079: 5382: 6197: 4201: 1888: 8672: 5880: 2862: 8819: 7428: 6943: 2554: 2420: 7919: 4302: 1465: 6877: 4550: 1163: 1708:), 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. 573:

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.

5961: 5190: 4403: 2096: 1513: 7664: 8321: 8951: 6254: 2476: 6139: 1611: 6724: 4852: 534: 3498: 8026: 3445: 3263: 6461: 3395: 9187: 9128: 9068: 9008: 3728: 2085:(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). 691:. 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 ( 8871: 2069:. 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". 7828: 6688:{\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}},} 6369: 5870: 5608: 4892: 3335: 2853: 2360: 718:, 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. 2625: 8732: 3647:), ...} 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 8241: 7732: 7533: 7473: 6719: 2219: 453:

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

1821: 1776: 8314: 6979: 3447:, 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 1982: 1855: 7770: 7965: 6290: 6283: 7560: 6083: 3958:{\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.} 1005: 6086: 1498: 887: 562: 8215: 8046: 7939: 7687: 4570: 1183: 4309: 6987: 916: 4016: 737:

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

5558:{\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 } 5305: 8478:, 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: 4129: 1438:

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

5293: 3196:{\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}}} 538:

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

10033: 6144: 4992:{\displaystyle \operatorname {P} (\left|{\overline {X}}_{n}-\mu \right|\geq \varepsilon )\leq {\frac {\sigma ^{2}}{n\varepsilon ^{2}}}.} 707:

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

5821:{\displaystyle \varphi _{{\overline {X}}_{n}}(t)=\left^{n}=\left^{n}\,\rightarrow \,e^{it\mu },\quad {\text{as}}\quad n\to \infty .} 581:

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

4805: 2783:{\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}}} 591:. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 474: 10522: 8737: 7324: 810:("the law of large numbers"). Thereafter, it was known under both names, but the "law of large numbers" is most frequently used. 9993: 9599:

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

8190:{\displaystyle \Pr(|S_{n}|\geq n\epsilon )\leq {\frac {1}{(n\epsilon )^{4}}}{\mathbb {E} }\leq {\frac {C}{\epsilon ^{4}n^{2}}},} 2503: 2369: 4231: 813:

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

10353: 10246: 9762: 9429: 5949:μ is a constant, which implies that convergence in distribution to μ and convergence in probability to μ are equivalent (see 4503: 1116: 9073:

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

1954:{\displaystyle {\overline {X}}_{n}\ {\overset {\text{a.s.}}{\longrightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 846:

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

7565: 5940:{\displaystyle {\overline {X}}_{n}\,{\overset {\mathcal {D}}{\rightarrow }}\,\mu \qquad {\text{for}}\qquad n\to \infty .} 3017:{\displaystyle E\left({\frac {2^{X}(-1)^{X}}{X}}\right)=\ \sum _{x=1}^{\infty }{\frac {2^{x}(-1)^{x}}{x}}2^{-x}=-\ln(2)} 8879: 6202: 2425: 10312: 10280: 9976: 9495: 9462: 9291: 9198: 6092: 461:

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

9368: 3027: 1185:) and no correlation between random variables. In that case, the variance of the average of n random variables is 10527: 8263: 5950: 1439: 201: 137: 9133:

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

7970: 3450: 249: 110: 3400: 3218: 10512: 10432: 9223: 6027:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 5256:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 4469:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 3714: 3348: 2182:{\displaystyle {\overline {X}}_{n}-\operatorname {E} {\big }\ {\overset {\text{a.s.}}{\longrightarrow }}\ 0,} 1579:{\displaystyle {\overline {X}}_{n}\ {\overset {P}{\rightarrow }}\ \mu \qquad {\textrm {when}}\ n\to \infty .} 1432: 776:. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his 695:<1) is infinite. One way to generate the Cauchy-distributed example is where the random numbers equal the 9781: 9142: 9078: 9018: 8958: 10517: 10233:, Lecture Notes in Physics, vol. 739, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 63–78, 9912: 8824: 10422: 9577:

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

8453:{\displaystyle {\frac {N_{n}(C)}{n}}\thickapprox p=P(X\in C)=\int _{a-h}^{a+h}f(x)\,dx\thickapprox 2hf(a)} 7775: 644:

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

10427: 9248: 5872: 5841: 5579: 4863: 3291: 2799: 2331: 2575: 1695:{\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=1.} 9723: 9238: 9228: 5179: 1502: 402: 388: 105: 17: 10490: 8677: 5835: 3516:. Since the width of the distribution of the average is not zero, it must have a positive lower bound 9213: 6801:{\displaystyle \Pr \left(\omega :|S_{n}(\omega )|\geq n\epsilon {\mbox{ infinitely often}}\right)=0.} 4857: 4088: 688: 554: 221: 9727: 8220: 7692: 7478: 7433: 6698: 3042: 10481: 8259: 8250:

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

2569: 826: 793: 280: 275: 164: 149: 4572:). The independence of the random variables implies no correlation between them, and we have that 1785: 1732: 9602: 9208: 8269: 6546:{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}\neq 0\right)=0,} 1877: 259: 130: 6948: 2066: 1860:

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

801: 10532: 9454: 9283: 3799:{\displaystyle \left\|f(x,\theta )\right\|\leq d(x)\quad {\text{for all}}\ \theta \in \Theta .} 3602: 2793: 154: 9966: 9487: 1826: 10463: 10040: 9685: 9203: 7737: 6451:{\displaystyle \Pr \left(\omega :\lim _{n\to \infty }{\frac {S_{n}(\omega )}{n}}=0\right)=1.} 1726: 1428: 1100: 568: 546: 398:

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

1421:. Large or infinite variance will make the convergence slower, but the LLN holds anyway. 8: 5285: 843: 730: 680: 676: 640: 609: 244: 186: 174: 169: 9930: 2422:

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

10402: 10384: 10226: 10189: 10148: 10065: 9825: 9673: 9634: 9542: 9402: 9332: 8475: 8200: 8031: 7924: 7672: 4555: 3968: 2628: 2295:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} <\infty .} 1494: 1168: 834: 648: 367: 231: 120: 60: 37: 3337:

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

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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

5289: 2632: 1104: 797: 769: 557:, the expected value is the theoretical probability of success, and the average of 115: 45: 2048:{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.} 10332: 9218: 8506:

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

788:) in 1713. He named this his "Golden Theorem" but it became generally known as " 10485: 10339:. Handbook of econometrics. Vol. IV. Elsevier Science. pp. 2111–2245. 9320: 6359:{\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=0\right)=1,} 4101: 2855:

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

4492: 3983: 2560:, since the convergence is not necessarily uniform on the set where it holds. 1464: 1058:{\displaystyle {\overline {X}}_{n}\to \mu \quad {\textrm {as}}\ n\to \infty .} 822: 10506: 10372: 9870: 9861: 9844: 9669: 9538: 9328: 2859:

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

542:) will approach 3.5, with the precision increasing as more dice are rolled. 9994:"A Note on the Weak Law of Large Numbers for Exchangeable Random Variables" 9233: 5400:, ... have the same characteristic function, so we will simply denote this 355: 306: 216: 100: 9617:

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 .

4386:{\displaystyle {\overline {X}}_{n}={\tfrac {1}{n}}(X_{1}+\cdots +X_{n}).} 4115: 652: 539: 462: 226: 67: 55: 2074: 358:

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

10398: 10184: 10167: 10152: 10128: 9820: 9803: 9638: 9546: 7314:{\displaystyle X_{i}^{3}X_{j},X_{i}^{2}X_{j}X_{k},X_{i}X_{j}X_{k}X_{l}} 7191:{\displaystyle {\mathbb {E} }={\mathbb {E} }\left={\mathbb {E} }\left.} 5178:

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

1779: 1501:(iid) samples from a random variable with finite mean, the sample mean 988:{\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})} 399: 84: 30: 10491:

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.),

10144: 10129:"An Analytic Technique to Prove Borel's Strong Law of Large Numbers" 9653: 9630: 5576:

These rules can be used to calculate the characteristic function of

4074:{\displaystyle {\frac {N_{n}(E)}{n}}\to p{\text{ as }}n\to \infty .} 350: 5377:{\displaystyle \varphi _{X}(t)=1+it\mu +o(t),\quad t\rightarrow 0.} 4498: 4108: 1414: 1111: 414: 211: 10439: 10389: 6192:{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}<\infty } 4196:{\displaystyle \Pr(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}.} 2572:

distributed random variable with parameter 1. The random variable

1095:) exists according to Lebesgue integration and is finite. It does 9749:. Springer Texts in Statistics. New York, NY: Springer New York. 8667:{\displaystyle (b-a){\tfrac {f(X_{1})+f(X_{2})+...+f(X_{n})}{n}}} 5412:

Among the basic properties of characteristic functions there are

1729:(normal distribution) with mean zero, but with variance equal to 696: 383: 10172:

Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete

9424:. Springer texts in statistics. London : Springer. p. 187. 6695:

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

4225: 3572:)=1 and the average will attain ε an infinite number of times.) 734: 700: 406: 10375:(2013). "A Tricentenary history of the Law of Large Numbers". 10096:"Asymptotic Properties of Non-Linear Least Squares Estimators" 9958: 9745:

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

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

4304:, we are interested in the convergence of the sample average 699:

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

410: 9931:"What Is the Law of Large Numbers? (Definition) | Built In" 9747:

A Course in Mathematical Statistics and Large Sample Theory

9744: 8814:{\displaystyle (b-a)\int _{a}^{b}f(x){\tfrac {1}{b-a}}{dx}} 7921:

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

7423:{\displaystyle {\mathbb {E} }={\mathbb {E} }{\mathbb {E} }} 6938:{\displaystyle \sum _{n=1}^{\infty }\Pr(A_{n})<\infty ,} 4493:

Proof using Chebyshev's inequality assuming finite variance

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states the conditions under which the convergence happens

2549:{\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } 2415:{\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon } 23:

Averages of repeated trials converge to the expected value

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Wiley Interdisciplinary Reviews: Computational Statistics

9384: 7914:{\displaystyle {\mathbb {E} }=n\tau +3n(n-1)\sigma ^{4}.} 4297:{\displaystyle E(X_{1})=E(X_{2})=\cdots =\mu <\infty } 1823:. The variance of the average is therefore asymptotic to 911:, both versions of the law state that the sample average 10307:] (in Danish) (3rd ed.). Copenhagen: HCØ-tryk. 10168:"An elementary proof of the strong law of large numbers" 9804:"An elementary proof of the strong law of large numbers" 6872:{\displaystyle A_{n}=\{\omega :|S_{n}|\geq n\epsilon \}} 3203:

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

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Although the proportion of heads (and tails) approaches

4545:{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} 1158:{\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} 8784: 8579: 6781: 6676: 5467: 4623: 4334: 3504:, there is probability which does not go to zero with 3453: 3403: 3269:

so that the denominator is positive) with probability

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

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will converge to the theoretical probability. For a

9965:Lehmann, Erich L.; Romano, Joseph P. (2006-03-30). 9482:

A Modern Introduction to Probability and Statistics

9449:

A Modern Introduction to Probability and Statistics

9278:

A Modern Introduction to Probability and Statistics

7659:{\displaystyle {\mathbb {E} }=({\mathbb {E} })^{2}} 5280:

Proof using convergence of characteristic functions

2316:

Differences between the weak law and the strong law

2075:

differences between the weak law and the strong law

9698: 9181: 9122: 9062: 9002: 8945: 8865: 8813: 8726: 8666: 8452: 8308: 8235: 8209: 8189: 8040: 8020: 7959: 7933: 7913: 7822: 7764: 7726: 7681: 7658: 7554: 7527: 7467: 7422: 7313: 7190: 6973: 6937: 6871: 6800: 6713: 6687: 6545: 6450: 6358: 6277: 6248: 6191: 6133: 6077: 6026: 5939: 5864: 5820: 5602: 5557: 5376: 5255: 5164: 4991: 4886: 4846: 4790: 4564: 4544: 4468: 4385: 4296: 4195: 4073: 3957: 3798: 3524:), which means there is a probability of at least 3492: 3439: 3389: 3329: 3257: 3195: 3016: 2847: 2782: 2619: 2548: 2470: 2414: 2354: 2294: 2181: 2081:then for the average to converge almost surely on 2047: 1953: 1849: 1815: 1770: 1694: 1578: 1403: 1177: 1157: 1057: 987: 545:It follows from the law of large numbers that the 528: 10270: 10001:Communications of the Korean Mathematical Society 9513:"Cauchy-Distributed Functions of Cauchy Variates" 6297: 1989: 1634: 1483:approaches 50% as the number of trials increases. 10504: 8946:{\displaystyle f(x)=cos^{2}(x){\sqrt {x^{3}+1}}} 8055: 6952: 6907: 6728: 6561: 6480: 6465: 6388: 6373: 6304: 6294: 6249:{\displaystyle {\mathbb {E} }=:\tau <\infty } 4133: 3823: 3577: 2471:{\displaystyle |{\overline {X}}_{n}-\mu |\neq 0} 1996: 1986: 1631: 1616: 1099:mean that the associated probability measure is 888:independent and identically distributed (i.i.d.) 563:independent and identically distributed (i.i.d.) 10298: 9658:Journal für die reine und angewandte Mathematik 9601:(in French). Paris, France: Bachelier. p. 9369:"The Law of Large Numbers and its Applications" 6134:{\displaystyle {\mathbb {E} }=:\mu <\infty } 3974: 601:. In particular, the proportion of heads after 442:results gets close to the expected value times 10337:Large sample estimation and hypothesis testing 9782:"The strong law of large numbers – What's new" 9690:: CS1 maint: DOI inactive as of August 2024 ( 9510: 10330: 10081: 9964: 2152: 2128: 858:that are described below. They are called the 331: 9590: 6866: 6828: 4847:{\displaystyle E({\overline {X}}_{n})=\mu .} 4228:random variables with finite expected value 2366:. Thus, it leaves open the possibility that 805: 529:{\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5} 10206: 9906: 9904: 8316:, for small positive h. Thus, for large n: 7198:We first claim that every term of the form 3556:, there is still a probability of at least 1427:of the random variables can be replaced by 777: 10271:Grimmett, G. R.; Stirzaker, D. R. (1992). 9910: 9776: 9774: 9718: 9716: 9651: 9589:Poisson names the "law of large numbers" ( 9511:Pitman, E. J. G.; Williams, E. J. (1967). 6614: 6610: 6053: 5001:This may be used to obtain the following: 3493:{\textstyle 1/{\sqrt {2\log \log \log n}}} 1497:'s law) states that given a collection of 338: 324: 10388: 10362: 10275:(2nd ed.). Oxford: Clarendon Press. 10227:"The Monte Carlo Method, an Introduction" 10183: 10111: 9860: 9819: 9722: 9579:(in Latin). Translated by Sheynin, Oscar. 9574: 9528: 8422: 8126: 8021:{\displaystyle {\mathbb {E} }\leq Cn^{2}} 7976: 7839: 7784: 7698: 7620: 7571: 7484: 7439: 7399: 7371: 7330: 7090: 7024: 6993: 6208: 6098: 5914: 5901: 5779: 5775: 4497:This proof uses the assumption of finite 3440:{\textstyle {\sqrt {k/\log \log \log k}}} 3258:{\textstyle {\sqrt {k/\log \log \log k}}} 2309: 1680: 1640: 10093: 9901: 9559: 5300:, with finite mean μ, can be written as 4205: 3536:trials. It will happen with probability 854:There are two different versions of the 761:), despite its underlying random nature. 725: 567: 349: 10478:Animations for the Law of Large Numbers 10289: 10165: 9842: 9808:Wahrscheinlichkeitstheorie Verw Gebiete 9801: 9771: 9713: 9616: 9596: 9477: 9444: 9419: 9273: 3532:) that the average will attain ε after 3508:, while the average sometime after the 3390:{\displaystyle \log n/\log \log \log n} 1876:'s law) states that the sample average 1499:independent and identically distributed 893:random variables with expected value E( 796:, named after Jacob Bernoulli's nephew 647:Another good example of the LLN is the 565:) is precisely the relative frequency. 10505: 10371: 10224: 9784:. Terrytao.wordpress.com. 19 June 2008 9707: 9306: 9182:{\displaystyle {\frac {e^{x}-1}{e-1}}} 9136:Another example is the integration of 9123:{\displaystyle \int _{-1}^{2}f(x){dx}} 9063:{\displaystyle \int _{-1}^{2}f(x){dx}} 9003:{\displaystyle \int _{-1}^{2}f(x){dx}} 8243:, we have established the Strong LLN. 4395:The weak law of large numbers states: 10459: 10440: 10321: 10220: 10218: 10100:The Annals of Mathematical Statistics 10031: 9895: 9883: 9564:. New York: Random House. p. 50. 9517:The Annals of Mathematical Statistics 8866:{\displaystyle \int _{a}^{b}f(x){dx}} 651:. These methods are a broad class of 10343: 10063: 9992:Dguvl Hun Hong; Sung Ho Lee (1998). 9953: 9362: 9360: 9358: 9356: 9354: 9352: 9350: 9348: 9346: 9269: 9267: 9265: 9263: 8247:Another proof was given by Etemadi. 7823:{\displaystyle ({\mathbb {E} })^{2}} 5955: 5184: 4397: 2090: 1882: 1507: 999: 804:further described it under the name 792:". This should not be confused with 10231:Computational Many-Particle Physics 10126: 8482:Simulate uniform random variables X 5865:{\displaystyle {\overline {X}}_{n}} 5603:{\displaystyle {\overline {X}}_{n}} 4887:{\displaystyle {\overline {X}}_{n}} 4013:trials, then with probability one, 3994:its probability of occurrence, and 3330:{\displaystyle k/\log \log \log k.} 2848:{\displaystyle 2^{X}(-1)^{X}X^{-1}} 2355:{\displaystyle {\overline {X}}_{n}} 13: 10365:Large sample methods in statistics 10301:Videregående Sandsynlighedsregning 10215: 9309:IEEE Transactions on Fuzzy Systems 6929: 6902: 6615: 6571: 6490: 6398: 6314: 6243: 6186: 6128: 6018: 5931: 5908: 5812: 5247: 5070: 5010: 4903: 4460: 4291: 4065: 3943: 3902: 3833: 3790: 2937: 2709: 2620:{\displaystyle \sin(X)e^{X}X^{-1}} 2324:states that for a specified large 2286: 2239: 2120: 2006: 1945: 1626: 1570: 1049: 561:such variables (assuming they are 14: 10544: 10415: 10363:Sen, P. K; Singer, J. M. (1993). 10133:The American Mathematical Monthly 9366: 9343: 9260: 9199:Asymptotic equipartition property 8429: 8353: 1602:That is, for any positive number 997:converges to the expected value: 886:, ... is an infinite sequence of 10292:Probability: Theory and Examples 10273:Probability and Random Processes 9843:Kingman, J. F. C. (April 1978). 8727:{\displaystyle (b-a)E(f(X_{1}))} 8048:sufficiently large. By Markov, 3265:(starting at sufficiently large 3028:cumulative distribution function 1472: 1463: 1454: 1088:means that the expected value E( 469:of the average of the rolls is: 44: 10348:(8th ed.). Prentice Hall. 10200: 10159: 10120: 10087: 10075: 10057: 10025: 9985: 9947: 9923: 9889: 9877: 9836: 9795: 9738: 9645: 9619:Journal of the History of Ideas 9610: 9583: 9568: 9214:Keynes' Treatise on Probability 8469: 8253: 6001: 5951:Convergence of random variables 5924: 5918: 5805: 5799: 5554: 5487: 5481: 5364: 5230: 4443: 3990:denotes the event in question, 3775: 2214:has a finite second moment and 1928: 1553: 1440:Convergence of random variables 1032: 10523:Asymptotic theory (statistics) 10294:(2nd ed.). Duxbury Press. 9968:Weak law converges to constant 9575:Bernoulli, Jakob (1713). "4". 9553: 9504: 9471: 9438: 9413: 9378: 9300: 9109: 9103: 9049: 9043: 8989: 8983: 8920: 8914: 8892: 8886: 8852: 8846: 8780: 8774: 8753: 8741: 8721: 8718: 8705: 8699: 8693: 8681: 8654: 8641: 8620: 8607: 8598: 8585: 8575: 8563: 8447: 8441: 8419: 8413: 8377: 8365: 8344: 8338: 8303: 8279: 8236:{\displaystyle \epsilon >0} 8149: 8131: 8112: 8102: 8090: 8077: 8062: 8058: 7999: 7981: 7895: 7883: 7862: 7844: 7811: 7807: 7789: 7779: 7759: 7747: 7727:{\displaystyle {\mathbb {E} }} 7721: 7703: 7647: 7643: 7625: 7615: 7609: 7576: 7528:{\displaystyle {\mathbb {E} }} 7522: 7489: 7468:{\displaystyle {\mathbb {E} }} 7462: 7444: 7417: 7404: 7394: 7376: 7363: 7335: 7016: 6998: 6968: 6955: 6923: 6910: 6853: 6838: 6767: 6763: 6757: 6743: 6714:{\displaystyle \epsilon >0} 6653: 6647: 6611: 6595: 6589: 6568: 6514: 6508: 6487: 6458:It is equivalent to show that 6422: 6416: 6395: 6311: 6231: 6213: 6167: 6154: 6116: 6103: 6015: 5987: 5928: 5904: 5809: 5776: 5658: 5652: 5551: 5545: 5532: 5526: 5510: 5504: 5478: 5463: 5447: 5441: 5368: 5358: 5352: 5325: 5319: 5244: 5216: 5118: 5076: 5058: 5016: 4951: 4909: 4832: 4812: 4730: 4698: 4669: 4666: 4634: 4619: 4607: 4587: 4526: 4513: 4457: 4429: 4377: 4345: 4273: 4260: 4251: 4238: 4224:, ... an infinite sequence of 4167: 4154: 4140: 4136: 4062: 4048: 4039: 4033: 3939: 3933: 3929: 3926: 3914: 3908: 3896: 3877: 3839: 3772: 3766: 3756: 3752: 3740: 3733: 3702:s, and measurable function of 3512:th trial will come back up to 3163: 3154: 3125: 3119: 3092: 3086: 3060: 3054: 3011: 3005: 2965: 2955: 2896: 2886: 2823: 2813: 2729: 2723: 2664: 2658: 2591: 2585: 2536: 2508: 2458: 2430: 2402: 2374: 2280: 2267: 2162: 2003: 1942: 1914: 1765: 1753: 1670: 1642: 1623: 1567: 1539: 1343: 1311: 1282: 1279: 1247: 1232: 1220: 1200: 1139: 1126: 1046: 1026: 982: 950: 687:becomes larger; the reason is 683:(α<1) will not converge as 111:Collectively exhaustive events 1: 10464:"Strong Law of Large Numbers" 10346:A first course in probability 10263: 10207:Billingsley, Patrick (1979). 9913:"Strong law of large numbers" 9420:Dekking, Michel, ed. (2005). 9224:Law of the iterated logarithm 7941:, and as such there exists a 3578:Uniform laws of large numbers 1863: 1435:in both versions of the law. 666: 10094:Jennrich, Robert I. (1969). 10007:(2): 385–391. Archived from 8876:We can find the integral of 8264:Borel's law of large numbers 6325: 5971: 5890: 5851: 5639: 5589: 5200: 5090: 5030: 4923: 4873: 4821: 4596: 4413: 4319: 3980:Borel's law of large numbers 3975:Borel's law of large numbers 3660:uniform law of large numbers 3584:uniform law of large numbers 2518: 2440: 2384: 2341: 2139: 2106: 2017: 1898: 1816:{\displaystyle n^{2}/\log n} 1771:{\displaystyle 2n/\log(n+1)} 1652: 1523: 1209: 1015: 926: 872:. Stated for the case where 430:The LLN only applies to the 7: 10445:"Weak Law of Large Numbers" 10428:Encyclopedia of Mathematics 10305:Advanced Probability Theory 10239:10.1007/978-3-540-74686-7_3 10166:Etemadi, Nasrollah (1981). 9917:Encyclopedia of Mathematics 9732:Encyclopedia of Mathematics 9249:Strong law of small numbers 9192: 8514:for i= 1, 2, ..., n. Then X 8309:{\displaystyle C=(a-h,a+h]} 6879:, and if we can show that 6040: 5269: 4482: 2556:holds for all large enough 2304:This statement is known as 2195: 1967: 1870:strong law of large numbers 1592: 1445: 1081:(Lebesgue integrability of 1071: 807:"la loi des grands nombres" 456: 10: 10549: 9321:10.1109/TFUZZ.2015.2466080 9239:Regression toward the mean 9229:Law of truly large numbers 6974:{\displaystyle \Pr(A_{n})} 5180:convergence in probability 3725:) such that E < ∞, and 3658:A particular example of a 3279:for each. The variance of 768:The Italian mathematician 721: 714:And if the trials embed a 549:of success in a series of 18:Law of truly large numbers 15: 10326:(4th ed.). Springer. 10290:Durrett, Richard (1995). 10082:Newey & McFadden 1994 9849:The Annals of Probability 9845:"Uses of Exchangeability" 9755:10.1007/978-1-4939-4032-5 9592:la loi des grands nombres 5873:converges in distribution 2067:pointwise ergodic theorem 1491:weak law of large numbers 555:Bernoulli random variable 10299:Martin Jacobsen (1992). 9670:10.1515/crll.1846.33.259 9478:Dekking, Michel (2005). 9445:Dekking, Michel (2005). 9274:Dekking, Michel (2005). 9254: 8260:probability distribution 5296:of any random variable, 3809:Then E is continuous in 3691:) is continuous at each 3030:of a random variable is 1850:{\displaystyle 1/\log n} 1503:converges in probability 849: 413:in a single spin of the 281:Law of total probability 276:Conditional independence 165:Exponential distribution 150:Probability distribution 16:Not to be confused with 10480:by Yihui Xie using the 10209:Probability and Measure 10113:10.1214/aoms/1177697731 9802:Etemadi, N. Z. (1981). 9672:(inactive 2024-08-02). 9652:Tchebichef, P. (1846). 9597:Poisson, S. D. (1837). 9530:10.1214/aoms/1177698885 9209:Infinite monkey theorem 8544:Take the average of f(X 7765:{\displaystyle 3n(n-1)} 6054:Proof of the strong law 5836:Lévy continuity theorem 5294:characteristic function 3609:∈ Θ, and continuous in 2631:, which is an improper 2306:Kolmogorov's strong law 1878:converges almost surely 1721:values goes to zero as 786:The Art of Conjecturing 675:results taken from the 405:. For example, while a 260:Conditional probability 10528:Theorems in statistics 10423:"Law of large numbers" 10344:Ross, Sheldon (2009). 10322:Loève, Michel (1977). 10066:"Law of large numbers" 10034:"Law of large numbers" 9898:, Chapter 17.3, p. 251 9862:10.1214/aop/1176995566 9728:"Law of large numbers" 9591: 9183: 9124: 9064: 9004: 8947: 8867: 8815: 8728: 8668: 8454: 8310: 8237: 8211: 8191: 8042: 8022: 7961: 7960:{\displaystyle C>0} 7935: 7915: 7824: 7766: 7728: 7683: 7660: 7556: 7529: 7469: 7424: 7315: 7192: 7059: 6975: 6939: 6906: 6873: 6802: 6783: infinitely often 6715: 6689: 6547: 6452: 6360: 6279: 6278:{\displaystyle \mu =0} 6250: 6193: 6135: 6079: 6028: 5941: 5866: 5822: 5604: 5559: 5378: 5257: 5166: 4993: 4888: 4858:Chebyshev's inequality 4848: 4792: 4566: 4546: 4470: 4387: 4298: 4197: 4089:Chebyshev's inequality 4075: 4005:) the number of times 3959: 3873: 3800: 3494: 3441: 3391: 3331: 3259: 3197: 3018: 2941: 2849: 2784: 2621: 2550: 2472: 2416: 2356: 2310:Sen & Singer (1993 2296: 2243: 2183: 2049: 1955: 1880:to the expected value 1851: 1817: 1772: 1696: 1580: 1505:to the expected value 1405: 1179: 1159: 1059: 989: 806: 778: 765: 658:that rely on repeated 574: 530: 391:converges to the true 363: 202:Continuous or discrete 155:Bernoulli distribution 10367:. Chapman & Hall. 10178:. Springer: 119–122. 9560:Mlodinow, L. (2008). 9486:. Springer. pp. 9453:. Springer. pp. 9282:. Springer. pp. 9204:Central limit theorem 9184: 9125: 9065: 9005: 8948: 8868: 8816: 8729: 8669: 8455: 8311: 8238: 8212: 8192: 8043: 8023: 7962: 7936: 7916: 7825: 7767: 7729: 7684: 7661: 7557: 7555:{\displaystyle X_{i}} 7530: 7470: 7425: 7316: 7193: 7039: 6976: 6940: 6886: 6874: 6803: 6716: 6690: 6548: 6453: 6361: 6280: 6251: 6194: 6136: 6080: 6078:{\displaystyle X_{i}} 6029: 5942: 5867: 5823: 5605: 5560: 5379: 5258: 5167: 4994: 4889: 4849: 4793: 4567: 4547: 4471: 4388: 4299: 4206:Proof of the weak law 4198: 4076: 3960: 3853: 3801: 3613:. Then for any fixed 3495: 3442: 3392: 3332: 3260: 3198: 3019: 2921: 2850: 2785: 2622: 2551: 2473: 2417: 2362:is likely to be near 2357: 2297: 2223: 2184: 2050: 1956: 1852: 1818: 1773: 1727:Gaussian distribution 1697: 1581: 1429:pairwise independence 1406: 1180: 1160: 1101:absolutely continuous 1060: 990: 794:Bernoulli's principle 729: 626:approaches infinity. 571: 547:empirical probability 531: 382:that states that the 353: 160:Binomial distribution 10513:Probability theorems 10324:Probability theory 1 9886:, Chapter 1.4, p. 14 9143: 9079: 9019: 8959: 8880: 8825: 8738: 8678: 8560: 8322: 8270: 8221: 8201: 8052: 8032: 7971: 7945: 7925: 7834: 7776: 7738: 7693: 7673: 7566: 7539: 7479: 7434: 7325: 7202: 6988: 6949: 6883: 6812: 6725: 6699: 6557: 6462: 6370: 6291: 6263: 6203: 6145: 6093: 6062: 5962: 5881: 5842: 5625: 5580: 5418: 5306: 5191: 5007: 4900: 4864: 4806: 4578: 4556: 4504: 4404: 4310: 4232: 4130: 4107:and finite non-zero 4017: 4009:occurs in the first 3819: 3729: 3451: 3401: 3349: 3292: 3219: 3034: 2863: 2800: 2639: 2576: 2504: 2426: 2370: 2332: 2220: 2097: 1983: 1889: 1827: 1786: 1733: 1612: 1514: 1191: 1169: 1117: 1006: 917: 856:law of large numbers 681:Pareto distributions 639:, almost surely the 475: 372:law of large numbers 286:Law of large numbers 255:Marginal probability 180:Poisson distribution 29:Part of a series on 10518:Mathematical proofs 10331:Newey, Whitney K.; 10064:J. Geyer, Charles. 9562:The Drunkard's Walk 9099: 9070:= 1.028 when n=250 9039: 8979: 8842: 8770: 8409: 8148: 7998: 7861: 7806: 7720: 7642: 7608: 7593: 7521: 7506: 7461: 7393: 7352: 7247: 7219: 7015: 6230: 5182:, we have obtained 3500:), but for a given 2713: 2312:, Theorem 2.3.10). 2205:provided that each 1425:Mutual independence 891:Lebesgue integrable 790:Bernoulli's theorem 677:Cauchy distribution 641:absolute difference 245:Complementary event 187:Probability measure 175:Pareto distribution 170:Normal distribution 10461:Weisstein, Eric W. 10442:Weisstein, Eric W. 10399:10.3150/12-BEJSP12 10185:10.1007/BF01013465 10032:Mukherjee, Sayan. 9821:10.1007/BF01013465 9179: 9120: 9082: 9060: 9022: 9010:= 0.905 when n=25 9000: 8962: 8943: 8863: 8828: 8811: 8801: 8756: 8724: 8664: 8662: 8476:Monte Carlo Method 8450: 8383: 8306: 8233: 8207: 8187: 8134: 8038: 8018: 7984: 7957: 7931: 7911: 7847: 7820: 7792: 7772:terms of the form 7762: 7724: 7706: 7689:terms of the form 7679: 7656: 7628: 7594: 7579: 7552: 7525: 7507: 7492: 7465: 7447: 7420: 7379: 7338: 7311: 7233: 7205: 7188: 7139: 7001: 6971: 6935: 6869: 6808:Define the events 6798: 6785: 6711: 6685: 6680: 6575: 6543: 6494: 6448: 6402: 6356: 6318: 6275: 6246: 6216: 6189: 6131: 6075: 6024: 5937: 5862: 5818: 5600: 5555: 5476: 5374: 5253: 5162: 4989: 4884: 4844: 4788: 4632: 4562: 4542: 4466: 4383: 4343: 4294: 4193: 4071: 3969:Extremum estimator 3955: 3837: 3796: 3490: 3437: 3387: 3327: 3255: 3193: 3188: 3014: 2845: 2780: 2693: 2629:Dirichlet integral 2617: 2546: 2468: 2412: 2352: 2292: 2179: 2045: 2010: 1951: 1857:and goes to zero. 1847: 1813: 1768: 1692: 1630: 1576: 1401: 1245: 1175: 1155: 1055: 985: 766: 711:goes to infinity. 649:Monte Carlo method 575: 526: 465:. Therefore, the 368:probability theory 364: 296:Boole's inequality 232:Stochastic process 121:Mutual exclusivity 38:Probability theory 10355:978-0-13-603313-4 10248:978-3-540-74685-0 10127:Wen, Liu (1991). 9764:978-1-4939-4030-1 9431:978-1-85233-896-1 9399:10.1002/wics.1314 9177: 8941: 8800: 8661: 8351: 8210:{\displaystyle n} 8182: 8122: 8041:{\displaystyle n} 7934:{\displaystyle n} 7682:{\displaystyle n} 7100: 6784: 6679: 6674: 6660: 6602: 6560: 6521: 6479: 6429: 6387: 6328: 6303: 6048: 6047: 6011: 6006: 5997: 5993: 5984: 5974: 5922: 5912: 5893: 5854: 5803: 5758: 5738: 5692: 5642: 5592: 5573:are independent. 5485: 5475: 5434: 5290:complex functions 5277: 5276: 5240: 5235: 5226: 5222: 5213: 5203: 5157: 5093: 5033: 4984: 4926: 4876: 4824: 4783: 4763: 4690: 4631: 4599: 4565:{\displaystyle i} 4490: 4489: 4453: 4448: 4439: 4435: 4426: 4416: 4342: 4322: 4188: 4057: 4046: 3951: 3947: 3851: 3822: 3783: 3779: 3552:. But even after 3548:which depends on 3488: 3435: 3345:is asymptotic to 3253: 3215:be plus or minus 3167: 3096: 2978: 2920: 2909: 2778: 2746: 2692: 2681: 2521: 2443: 2387: 2344: 2259: 2203: 2202: 2172: 2168: 2167: 2159: 2142: 2109: 2020: 1995: 1975: 1974: 1938: 1933: 1924: 1920: 1919: 1911: 1901: 1655: 1615: 1600: 1599: 1563: 1558: 1549: 1545: 1536: 1526: 1396: 1376: 1303: 1244: 1212: 1178:{\displaystyle i} 1079: 1078: 1042: 1037: 1018: 948: 929: 518: 425:gambler's fallacy 348: 347: 250:Joint probability 197:Bernoulli process 96:Probability space 10540: 10497:Business Insider 10474: 10473: 10455: 10454: 10436: 10410: 10392: 10383:(4): 1088–1121. 10368: 10359: 10340: 10333:McFadden, Daniel 10327: 10318: 10295: 10286: 10258: 10257: 10256: 10255: 10222: 10213: 10212: 10204: 10198: 10197: 10187: 10163: 10157: 10156: 10124: 10118: 10117: 10115: 10091: 10085: 10079: 10073: 10072: 10070: 10061: 10055: 10054: 10052: 10051: 10045: 10039:. Archived from 10038: 10029: 10023: 10022: 10020: 10019: 10013: 9998: 9989: 9983: 9982: 9962: 9956: 9951: 9945: 9944: 9942: 9941: 9927: 9921: 9920: 9911:Yuri Prokhorov. 9908: 9899: 9893: 9887: 9881: 9875: 9874: 9864: 9840: 9834: 9833: 9823: 9799: 9793: 9792: 9790: 9789: 9778: 9769: 9768: 9742: 9736: 9735: 9720: 9711: 9705: 9696: 9695: 9689: 9681: 9649: 9643: 9642: 9614: 9608: 9606: 9594: 9587: 9581: 9580: 9572: 9566: 9565: 9557: 9551: 9550: 9532: 9508: 9502: 9501: 9485: 9475: 9469: 9468: 9452: 9442: 9436: 9435: 9417: 9411: 9410: 9382: 9376: 9375: 9373: 9364: 9341: 9340: 9304: 9298: 9297: 9281: 9271: 9188: 9186: 9185: 9180: 9178: 9176: 9165: 9158: 9157: 9147: 9129: 9127: 9126: 9121: 9119: 9098: 9093: 9069: 9067: 9066: 9061: 9059: 9038: 9033: 9009: 9007: 9006: 9001: 8999: 8978: 8973: 8952: 8950: 8949: 8944: 8942: 8934: 8933: 8924: 8913: 8912: 8872: 8870: 8869: 8864: 8862: 8841: 8836: 8820: 8818: 8817: 8812: 8810: 8802: 8799: 8785: 8769: 8764: 8733: 8731: 8730: 8725: 8717: 8716: 8673: 8671: 8670: 8665: 8663: 8657: 8653: 8652: 8619: 8618: 8597: 8596: 8580: 8459: 8457: 8456: 8451: 8408: 8397: 8352: 8347: 8337: 8336: 8326: 8315: 8313: 8312: 8307: 8242: 8240: 8239: 8234: 8216: 8214: 8213: 8208: 8196: 8194: 8193: 8188: 8183: 8181: 8180: 8179: 8170: 8169: 8156: 8147: 8142: 8130: 8129: 8123: 8121: 8120: 8119: 8097: 8080: 8075: 8074: 8065: 8047: 8045: 8044: 8039: 8027: 8025: 8024: 8019: 8017: 8016: 7997: 7992: 7980: 7979: 7966: 7964: 7963: 7958: 7940: 7938: 7937: 7932: 7920: 7918: 7917: 7912: 7907: 7906: 7860: 7855: 7843: 7842: 7829: 7827: 7826: 7821: 7819: 7818: 7805: 7800: 7788: 7787: 7771: 7769: 7768: 7763: 7733: 7731: 7730: 7725: 7719: 7714: 7702: 7701: 7688: 7686: 7685: 7680: 7665: 7663: 7662: 7657: 7655: 7654: 7641: 7636: 7624: 7623: 7607: 7602: 7592: 7587: 7575: 7574: 7561: 7559: 7558: 7553: 7551: 7550: 7534: 7532: 7531: 7526: 7520: 7515: 7505: 7500: 7488: 7487: 7474: 7472: 7471: 7466: 7460: 7455: 7443: 7442: 7429: 7427: 7426: 7421: 7416: 7415: 7403: 7402: 7392: 7387: 7375: 7374: 7362: 7361: 7351: 7346: 7334: 7333: 7320: 7318: 7317: 7312: 7310: 7309: 7300: 7299: 7290: 7289: 7280: 7279: 7267: 7266: 7257: 7256: 7246: 7241: 7229: 7228: 7218: 7213: 7197: 7195: 7194: 7189: 7184: 7180: 7179: 7178: 7169: 7168: 7159: 7158: 7149: 7148: 7138: 7094: 7093: 7084: 7080: 7079: 7074: 7070: 7069: 7068: 7058: 7053: 7028: 7027: 7014: 7009: 6997: 6996: 6980: 6978: 6977: 6972: 6967: 6966: 6944: 6942: 6941: 6936: 6922: 6921: 6905: 6900: 6878: 6876: 6875: 6870: 6856: 6851: 6850: 6841: 6824: 6823: 6807: 6805: 6804: 6799: 6791: 6787: 6786: 6782: 6770: 6756: 6755: 6746: 6720: 6718: 6717: 6712: 6694: 6692: 6691: 6686: 6681: 6678:infinitely often 6677: 6672: 6665: 6661: 6656: 6646: 6645: 6635: 6603: 6598: 6588: 6587: 6577: 6574: 6552: 6550: 6549: 6544: 6533: 6529: 6522: 6517: 6507: 6506: 6496: 6493: 6457: 6455: 6454: 6449: 6441: 6437: 6430: 6425: 6415: 6414: 6404: 6401: 6365: 6363: 6362: 6357: 6346: 6342: 6335: 6334: 6329: 6321: 6317: 6284: 6282: 6281: 6276: 6255: 6253: 6252: 6247: 6229: 6224: 6212: 6211: 6198: 6196: 6195: 6190: 6182: 6181: 6166: 6165: 6140: 6138: 6137: 6132: 6115: 6114: 6102: 6101: 6084: 6082: 6081: 6076: 6074: 6073: 6042: 6033: 6031: 6030: 6025: 6009: 6008: 6007: 6004: 5995: 5994: 5986: 5982: 5981: 5980: 5975: 5967: 5956: 5946: 5944: 5943: 5938: 5923: 5920: 5913: 5911: 5903: 5900: 5899: 5894: 5886: 5871: 5869: 5868: 5863: 5861: 5860: 5855: 5847: 5827: 5825: 5824: 5819: 5804: 5801: 5795: 5794: 5774: 5773: 5768: 5764: 5763: 5759: 5751: 5739: 5731: 5708: 5707: 5702: 5698: 5697: 5693: 5685: 5679: 5678: 5651: 5650: 5649: 5648: 5643: 5635: 5609: 5607: 5606: 5601: 5599: 5598: 5593: 5585: 5564: 5562: 5561: 5556: 5544: 5543: 5525: 5524: 5503: 5502: 5486: 5483: 5477: 5468: 5462: 5461: 5440: 5439: 5435: 5427: 5383: 5381: 5380: 5375: 5318: 5317: 5286:Taylor's theorem 5271: 5262: 5260: 5259: 5254: 5238: 5237: 5236: 5233: 5224: 5223: 5215: 5211: 5210: 5209: 5204: 5196: 5185: 5171: 5169: 5168: 5163: 5158: 5156: 5155: 5154: 5141: 5140: 5131: 5111: 5107: 5100: 5099: 5094: 5086: 5051: 5047: 5040: 5039: 5034: 5026: 4998: 4996: 4995: 4990: 4985: 4983: 4982: 4981: 4968: 4967: 4958: 4944: 4940: 4933: 4932: 4927: 4919: 4893: 4891: 4890: 4885: 4883: 4882: 4877: 4869: 4853: 4851: 4850: 4845: 4831: 4830: 4825: 4817: 4797: 4795: 4794: 4789: 4784: 4779: 4778: 4769: 4764: 4762: 4761: 4752: 4751: 4750: 4737: 4729: 4728: 4710: 4709: 4691: 4689: 4688: 4676: 4665: 4664: 4646: 4645: 4633: 4624: 4606: 4605: 4600: 4592: 4571: 4569: 4568: 4563: 4551: 4549: 4548: 4543: 4541: 4540: 4525: 4524: 4484: 4475: 4473: 4472: 4467: 4451: 4450: 4449: 4446: 4437: 4436: 4428: 4424: 4423: 4422: 4417: 4409: 4398: 4392: 4390: 4389: 4384: 4376: 4375: 4357: 4356: 4344: 4335: 4329: 4328: 4323: 4315: 4303: 4301: 4300: 4295: 4272: 4271: 4250: 4249: 4202: 4200: 4199: 4194: 4189: 4187: 4186: 4174: 4157: 4143: 4123: 4080: 4078: 4077: 4072: 4058: 4055: 4047: 4042: 4032: 4031: 4021: 3964: 3962: 3961: 3956: 3949: 3948: 3946: 3938: 3936: 3932: 3889: 3888: 3872: 3867: 3852: 3844: 3836: 3805: 3803: 3802: 3797: 3781: 3780: 3777: 3759: 3755: 3617:, the sequence { 3544:)/2 before some 3499: 3497: 3496: 3491: 3489: 3463: 3461: 3446: 3444: 3443: 3438: 3436: 3413: 3405: 3396: 3394: 3393: 3388: 3365: 3336: 3334: 3333: 3328: 3302: 3278: 3277: 3273: 3264: 3262: 3261: 3256: 3254: 3231: 3223: 3202: 3200: 3199: 3194: 3192: 3191: 3168: 3166: 3134: 3097: 3095: 3069: 3023: 3021: 3020: 3015: 2992: 2991: 2979: 2974: 2973: 2972: 2954: 2953: 2943: 2940: 2935: 2918: 2914: 2910: 2905: 2904: 2903: 2885: 2884: 2874: 2854: 2852: 2851: 2846: 2844: 2843: 2831: 2830: 2812: 2811: 2789: 2787: 2786: 2781: 2779: 2771: 2760: 2759: 2747: 2742: 2741: 2740: 2715: 2712: 2707: 2690: 2686: 2682: 2677: 2676: 2675: 2650: 2633:Riemann integral 2626: 2624: 2623: 2618: 2616: 2615: 2603: 2602: 2555: 2553: 2552: 2547: 2539: 2528: 2527: 2522: 2514: 2511: 2499: 2489:shows that this 2477: 2475: 2474: 2469: 2461: 2450: 2449: 2444: 2436: 2433: 2421: 2419: 2418: 2413: 2405: 2394: 2393: 2388: 2380: 2377: 2361: 2359: 2358: 2353: 2351: 2350: 2345: 2337: 2301: 2299: 2298: 2293: 2279: 2278: 2260: 2258: 2257: 2245: 2242: 2237: 2197: 2188: 2186: 2185: 2180: 2170: 2169: 2165: 2161: 2157: 2156: 2155: 2149: 2148: 2143: 2135: 2132: 2131: 2116: 2115: 2110: 2102: 2091: 2054: 2052: 2051: 2046: 2038: 2034: 2027: 2026: 2021: 2013: 2009: 1969: 1960: 1958: 1957: 1952: 1936: 1935: 1934: 1931: 1922: 1921: 1917: 1913: 1909: 1908: 1907: 1902: 1894: 1883: 1856: 1854: 1853: 1848: 1837: 1822: 1820: 1819: 1814: 1803: 1798: 1797: 1777: 1775: 1774: 1769: 1746: 1701: 1699: 1698: 1693: 1685: 1681: 1673: 1662: 1661: 1656: 1648: 1645: 1629: 1594: 1585: 1583: 1582: 1577: 1561: 1560: 1559: 1556: 1547: 1546: 1538: 1534: 1533: 1532: 1527: 1519: 1508: 1476: 1467: 1458: 1410: 1408: 1407: 1402: 1397: 1392: 1391: 1382: 1377: 1375: 1374: 1365: 1364: 1363: 1350: 1342: 1341: 1323: 1322: 1304: 1302: 1301: 1289: 1278: 1277: 1259: 1258: 1246: 1237: 1219: 1218: 1213: 1205: 1184: 1182: 1181: 1176: 1164: 1162: 1161: 1156: 1154: 1153: 1138: 1137: 1105:Lebesgue measure 1103:with respect to 1073: 1064: 1062: 1061: 1056: 1040: 1039: 1038: 1035: 1025: 1024: 1019: 1011: 1000: 994: 992: 991: 986: 981: 980: 962: 961: 949: 941: 936: 935: 930: 922: 870:of large numbers 863:of large numbers 809: 798:Daniel Bernoulli 783: 770:Gerolamo Cardano 757:phenomenon (see 638: 637: 633: 621: 620: 616: 600: 599: 595: 590: 589: 585: 551:Bernoulli trials 535: 533: 532: 527: 519: 514: 479: 380:mathematical law 340: 333: 326: 116:Elementary event 48: 26: 25: 10548: 10547: 10543: 10542: 10541: 10539: 10538: 10537: 10503: 10502: 10421: 10418: 10413: 10356: 10315: 10283: 10266: 10261: 10253: 10251: 10249: 10223: 10216: 10205: 10201: 10164: 10160: 10145:10.2307/2323947 10125: 10121: 10092: 10088: 10080: 10076: 10068: 10062: 10058: 10049: 10047: 10043: 10036: 10030: 10026: 10017: 10015: 10011: 9996: 9990: 9986: 9979: 9963: 9959: 9952: 9948: 9939: 9937: 9929: 9928: 9924: 9909: 9902: 9894: 9890: 9882: 9878: 9841: 9837: 9800: 9796: 9787: 9785: 9780: 9779: 9772: 9765: 9743: 9739: 9721: 9714: 9706: 9699: 9683: 9682: 9664:(33): 259–267. 9650: 9646: 9631:10.2307/2709176 9615: 9611: 9588: 9584: 9573: 9569: 9558: 9554: 9509: 9505: 9498: 9476: 9472: 9465: 9443: 9439: 9432: 9418: 9414: 9383: 9379: 9371: 9365: 9344: 9305: 9301: 9294: 9272: 9261: 9257: 9219:Law of averages 9195: 9166: 9153: 9149: 9148: 9146: 9144: 9141: 9140: 9139: 9112: 9094: 9086: 9080: 9077: 9076: 9052: 9034: 9026: 9020: 9017: 9016: 8992: 8974: 8966: 8960: 8957: 8956: 8929: 8925: 8923: 8908: 8904: 8881: 8878: 8877: 8855: 8837: 8832: 8826: 8823: 8822: 8803: 8789: 8783: 8765: 8760: 8739: 8736: 8735: 8712: 8708: 8679: 8676: 8675: 8648: 8644: 8614: 8610: 8592: 8588: 8581: 8578: 8561: 8558: 8557: 8556:) by computing 8555: 8551: 8547: 8540: 8536: 8532: 8525: 8521: 8517: 8513: 8509: 8505: 8501: 8497: 8493: 8489: 8485: 8472: 8398: 8387: 8332: 8328: 8327: 8325: 8323: 8320: 8319: 8271: 8268: 8267: 8256: 8246: 8222: 8219: 8218: 8202: 8199: 8198: 8175: 8171: 8165: 8161: 8160: 8155: 8143: 8138: 8125: 8124: 8115: 8111: 8101: 8096: 8076: 8070: 8066: 8061: 8053: 8050: 8049: 8033: 8030: 8029: 8012: 8008: 7993: 7988: 7975: 7974: 7972: 7969: 7968: 7946: 7943: 7942: 7926: 7923: 7922: 7902: 7898: 7856: 7851: 7838: 7837: 7835: 7832: 7831: 7814: 7810: 7801: 7796: 7783: 7782: 7777: 7774: 7773: 7739: 7736: 7735: 7715: 7710: 7697: 7696: 7694: 7691: 7690: 7674: 7671: 7670: 7650: 7646: 7637: 7632: 7619: 7618: 7603: 7598: 7588: 7583: 7570: 7569: 7567: 7564: 7563: 7546: 7542: 7540: 7537: 7536: 7516: 7511: 7501: 7496: 7483: 7482: 7480: 7477: 7476: 7456: 7451: 7438: 7437: 7435: 7432: 7431: 7411: 7407: 7398: 7397: 7388: 7383: 7370: 7369: 7357: 7353: 7347: 7342: 7329: 7328: 7326: 7323: 7322: 7305: 7301: 7295: 7291: 7285: 7281: 7275: 7271: 7262: 7258: 7252: 7248: 7242: 7237: 7224: 7220: 7214: 7209: 7203: 7200: 7199: 7174: 7170: 7164: 7160: 7154: 7150: 7144: 7140: 7104: 7099: 7095: 7089: 7088: 7075: 7064: 7060: 7054: 7043: 7038: 7034: 7033: 7029: 7023: 7022: 7010: 7005: 6992: 6991: 6989: 6986: 6985: 6962: 6958: 6950: 6947: 6946: 6917: 6913: 6901: 6890: 6884: 6881: 6880: 6852: 6846: 6842: 6837: 6819: 6815: 6813: 6810: 6809: 6780: 6766: 6751: 6747: 6742: 6735: 6731: 6726: 6723: 6722: 6700: 6697: 6696: 6675: 6641: 6637: 6636: 6634: 6630: 6583: 6579: 6578: 6576: 6564: 6558: 6555: 6554: 6502: 6498: 6497: 6495: 6483: 6472: 6468: 6463: 6460: 6459: 6410: 6406: 6405: 6403: 6391: 6380: 6376: 6371: 6368: 6367: 6330: 6320: 6319: 6307: 6302: 6298: 6292: 6289: 6288: 6264: 6261: 6260: 6225: 6220: 6207: 6206: 6204: 6201: 6200: 6177: 6173: 6161: 6157: 6146: 6143: 6142: 6110: 6106: 6097: 6096: 6094: 6091: 6090: 6069: 6065: 6063: 6060: 6059: 6056: 6003: 6002: 5985: 5976: 5966: 5965: 5963: 5960: 5959: 5919: 5907: 5902: 5895: 5885: 5884: 5882: 5879: 5878: 5856: 5846: 5845: 5843: 5840: 5839: 5800: 5784: 5780: 5769: 5750: 5746: 5730: 5717: 5713: 5712: 5703: 5684: 5680: 5674: 5670: 5669: 5665: 5664: 5644: 5634: 5633: 5632: 5628: 5626: 5623: 5622: 5618: 5594: 5584: 5583: 5581: 5578: 5577: 5539: 5535: 5520: 5516: 5492: 5488: 5482: 5466: 5457: 5453: 5426: 5425: 5421: 5419: 5416: 5415: 5408: 5399: 5392: 5313: 5309: 5307: 5304: 5303: 5282: 5232: 5231: 5214: 5205: 5195: 5194: 5192: 5189: 5188: 5150: 5146: 5142: 5136: 5132: 5130: 5095: 5085: 5084: 5083: 5079: 5035: 5025: 5024: 5023: 5019: 5008: 5005: 5004: 4977: 4973: 4969: 4963: 4959: 4957: 4928: 4918: 4917: 4916: 4912: 4901: 4898: 4897: 4878: 4868: 4867: 4865: 4862: 4861: 4826: 4816: 4815: 4807: 4804: 4803: 4774: 4770: 4768: 4757: 4753: 4746: 4742: 4738: 4736: 4724: 4720: 4705: 4701: 4684: 4680: 4675: 4660: 4656: 4641: 4637: 4622: 4601: 4591: 4590: 4579: 4576: 4575: 4557: 4554: 4553: 4536: 4532: 4520: 4516: 4505: 4502: 4501: 4495: 4445: 4444: 4427: 4418: 4408: 4407: 4405: 4402: 4401: 4371: 4367: 4352: 4348: 4333: 4324: 4314: 4313: 4311: 4308: 4307: 4267: 4263: 4245: 4241: 4233: 4230: 4229: 4223: 4216: 4208: 4182: 4178: 4173: 4153: 4139: 4131: 4128: 4127: 4118: 4114:. Then for any 4098:random variable 4054: 4027: 4023: 4022: 4020: 4018: 4015: 4014: 3999: 3977: 3942: 3937: 3884: 3880: 3868: 3857: 3843: 3842: 3838: 3826: 3820: 3817: 3816: 3776: 3736: 3732: 3730: 3727: 3726: 3713:there exists a 3655:) convergence. 3642: 3627: 3580: 3575: 3462: 3457: 3452: 3449: 3448: 3409: 3404: 3402: 3399: 3398: 3361: 3350: 3347: 3346: 3298: 3293: 3290: 3289: 3287: 3275: 3271: 3270: 3227: 3222: 3220: 3217: 3216: 3214: 3187: 3186: 3172: 3138: 3133: 3128: 3113: 3112: 3101: 3073: 3068: 3063: 3038: 3037: 3035: 3032: 3031: 2984: 2980: 2968: 2964: 2949: 2945: 2944: 2942: 2936: 2925: 2899: 2895: 2880: 2876: 2875: 2873: 2869: 2864: 2861: 2860: 2836: 2832: 2826: 2822: 2807: 2803: 2801: 2798: 2797: 2770: 2752: 2748: 2736: 2732: 2716: 2714: 2708: 2697: 2671: 2667: 2651: 2649: 2645: 2640: 2637: 2636: 2608: 2604: 2598: 2594: 2577: 2574: 2573: 2535: 2523: 2513: 2512: 2507: 2505: 2502: 2501: 2500:the inequality 2494: 2457: 2445: 2435: 2434: 2429: 2427: 2424: 2423: 2401: 2389: 2379: 2378: 2373: 2371: 2368: 2367: 2346: 2336: 2335: 2333: 2330: 2329: 2318: 2274: 2270: 2253: 2249: 2244: 2238: 2227: 2221: 2218: 2217: 2213: 2160: 2151: 2150: 2144: 2134: 2133: 2127: 2126: 2111: 2101: 2100: 2098: 2095: 2094: 2022: 2012: 2011: 1999: 1994: 1990: 1984: 1981: 1980: 1930: 1929: 1912: 1903: 1893: 1892: 1890: 1887: 1886: 1866: 1833: 1828: 1825: 1824: 1799: 1793: 1789: 1787: 1784: 1783: 1742: 1734: 1731: 1730: 1669: 1657: 1647: 1646: 1641: 1639: 1635: 1619: 1613: 1610: 1609: 1555: 1554: 1537: 1528: 1518: 1517: 1515: 1512: 1511: 1487: 1486: 1485: 1484: 1479: 1478: 1477: 1469: 1468: 1460: 1459: 1448: 1433:exchangeability 1387: 1383: 1381: 1370: 1366: 1359: 1355: 1351: 1349: 1337: 1333: 1318: 1314: 1297: 1293: 1288: 1273: 1269: 1254: 1250: 1235: 1214: 1204: 1203: 1192: 1189: 1188: 1170: 1167: 1166: 1149: 1145: 1133: 1129: 1118: 1115: 1114: 1093: 1086: 1034: 1033: 1020: 1010: 1009: 1007: 1004: 1003: 976: 972: 957: 953: 940: 931: 921: 920: 918: 915: 914: 906: 899: 885: 878: 852: 780:Ars Conjectandi 774:Jacob Bernoulli 764: 724: 669: 660:random sampling 635: 631: 630: 618: 614: 613: 597: 593: 592: 587: 583: 582: 577:For example, a 480: 478: 476: 473: 472: 459: 344: 192:Random variable 143:Bernoulli trial 24: 21: 12: 11: 5: 10546: 10536: 10535: 10530: 10525: 10520: 10515: 10501: 10500: 10488: 10475: 10456: 10437: 10417: 10416:External links 10414: 10412: 10411: 10373:Seneta, Eugene 10369: 10360: 10354: 10341: 10335:(1994). "36". 10328: 10319: 10313: 10296: 10287: 10281: 10267: 10265: 10262: 10260: 10259: 10247: 10214: 10199: 10158: 10139:(2): 146–148. 10119: 10106:(2): 633–643. 10086: 10074: 10056: 10024: 9984: 9977: 9957: 9946: 9922: 9900: 9888: 9876: 9835: 9814:(1): 119–122. 9794: 9770: 9763: 9737: 9712: 9697: 9644: 9625:(3): 455–475. 9609: 9582: 9567: 9552: 9523:(3): 916–918. 9503: 9496: 9470: 9463: 9437: 9430: 9412: 9393:(6): 386–392. 9377: 9367:Sedor, Kelly. 9342: 9315:(3): 615–621. 9299: 9292: 9258: 9256: 9253: 9252: 9251: 9246: 9241: 9236: 9231: 9226: 9221: 9216: 9211: 9206: 9201: 9194: 9191: 9175: 9172: 9169: 9164: 9161: 9156: 9152: 9137: 9118: 9115: 9111: 9108: 9105: 9102: 9097: 9092: 9089: 9085: 9058: 9055: 9051: 9048: 9045: 9042: 9037: 9032: 9029: 9025: 8998: 8995: 8991: 8988: 8985: 8982: 8977: 8972: 8969: 8965: 8940: 8937: 8932: 8928: 8922: 8919: 8916: 8911: 8907: 8903: 8900: 8897: 8894: 8891: 8888: 8885: 8874: 8873: 8861: 8858: 8854: 8851: 8848: 8845: 8840: 8835: 8831: 8809: 8806: 8798: 8795: 8792: 8788: 8782: 8779: 8776: 8773: 8768: 8763: 8759: 8755: 8752: 8749: 8746: 8743: 8723: 8720: 8715: 8711: 8707: 8704: 8701: 8698: 8695: 8692: 8689: 8686: 8683: 8660: 8656: 8651: 8647: 8643: 8640: 8637: 8634: 8631: 8628: 8625: 8622: 8617: 8613: 8609: 8606: 8603: 8600: 8595: 8591: 8587: 8584: 8577: 8574: 8571: 8568: 8565: 8553: 8549: 8545: 8542: 8538: 8534: 8530: 8527: 8523: 8519: 8515: 8511: 8507: 8503: 8499: 8495: 8491: 8487: 8483: 8471: 8468: 8449: 8446: 8443: 8440: 8437: 8434: 8431: 8428: 8425: 8421: 8418: 8415: 8412: 8407: 8404: 8401: 8396: 8393: 8390: 8386: 8382: 8379: 8376: 8373: 8370: 8367: 8364: 8361: 8358: 8355: 8350: 8346: 8343: 8340: 8335: 8331: 8305: 8302: 8299: 8296: 8293: 8290: 8287: 8284: 8281: 8278: 8275: 8262:. By applying 8255: 8252: 8232: 8229: 8226: 8206: 8186: 8178: 8174: 8168: 8164: 8159: 8154: 8151: 8146: 8141: 8137: 8133: 8128: 8118: 8114: 8110: 8107: 8104: 8100: 8095: 8092: 8089: 8086: 8083: 8079: 8073: 8069: 8064: 8060: 8057: 8037: 8015: 8011: 8007: 8004: 8001: 7996: 7991: 7987: 7983: 7978: 7956: 7953: 7950: 7930: 7910: 7905: 7901: 7897: 7894: 7891: 7888: 7885: 7882: 7879: 7876: 7873: 7870: 7867: 7864: 7859: 7854: 7850: 7846: 7841: 7817: 7813: 7809: 7804: 7799: 7795: 7791: 7786: 7781: 7761: 7758: 7755: 7752: 7749: 7746: 7743: 7723: 7718: 7713: 7709: 7705: 7700: 7678: 7653: 7649: 7645: 7640: 7635: 7631: 7627: 7622: 7617: 7614: 7611: 7606: 7601: 7597: 7591: 7586: 7582: 7578: 7573: 7549: 7545: 7524: 7519: 7514: 7510: 7504: 7499: 7495: 7491: 7486: 7464: 7459: 7454: 7450: 7446: 7441: 7419: 7414: 7410: 7406: 7401: 7396: 7391: 7386: 7382: 7378: 7373: 7368: 7365: 7360: 7356: 7350: 7345: 7341: 7337: 7332: 7308: 7304: 7298: 7294: 7288: 7284: 7278: 7274: 7270: 7265: 7261: 7255: 7251: 7245: 7240: 7236: 7232: 7227: 7223: 7217: 7212: 7208: 7187: 7183: 7177: 7173: 7167: 7163: 7157: 7153: 7147: 7143: 7137: 7134: 7131: 7128: 7125: 7122: 7119: 7116: 7113: 7110: 7107: 7103: 7098: 7092: 7087: 7083: 7078: 7073: 7067: 7063: 7057: 7052: 7049: 7046: 7042: 7037: 7032: 7026: 7021: 7018: 7013: 7008: 7004: 7000: 6995: 6970: 6965: 6961: 6957: 6954: 6934: 6931: 6928: 6925: 6920: 6916: 6912: 6909: 6904: 6899: 6896: 6893: 6889: 6868: 6865: 6862: 6859: 6855: 6849: 6845: 6840: 6836: 6833: 6830: 6827: 6822: 6818: 6797: 6794: 6790: 6779: 6776: 6773: 6769: 6765: 6762: 6759: 6754: 6750: 6745: 6741: 6738: 6734: 6730: 6710: 6707: 6704: 6684: 6671: 6668: 6664: 6659: 6655: 6652: 6649: 6644: 6640: 6633: 6629: 6626: 6623: 6620: 6617: 6613: 6609: 6606: 6601: 6597: 6594: 6591: 6586: 6582: 6573: 6570: 6567: 6563: 6542: 6539: 6536: 6532: 6528: 6525: 6520: 6516: 6513: 6510: 6505: 6501: 6492: 6489: 6486: 6482: 6478: 6475: 6471: 6467: 6447: 6444: 6440: 6436: 6433: 6428: 6424: 6421: 6418: 6413: 6409: 6400: 6397: 6394: 6390: 6386: 6383: 6379: 6375: 6355: 6352: 6349: 6345: 6341: 6338: 6333: 6327: 6324: 6316: 6313: 6310: 6306: 6301: 6296: 6274: 6271: 6268: 6245: 6242: 6239: 6236: 6233: 6228: 6223: 6219: 6215: 6210: 6188: 6185: 6180: 6176: 6172: 6169: 6164: 6160: 6156: 6153: 6150: 6130: 6127: 6124: 6121: 6118: 6113: 6109: 6105: 6100: 6072: 6068: 6055: 6052: 6046: 6045: 6036: 6034: 6023: 6020: 6017: 6014: 6000: 5992: 5989: 5979: 5973: 5970: 5953:.) Therefore, 5936: 5933: 5930: 5927: 5917: 5910: 5906: 5898: 5892: 5889: 5859: 5853: 5850: 5817: 5814: 5811: 5808: 5798: 5793: 5790: 5787: 5783: 5778: 5772: 5767: 5762: 5757: 5754: 5749: 5745: 5742: 5737: 5734: 5729: 5726: 5723: 5720: 5716: 5711: 5706: 5701: 5696: 5691: 5688: 5683: 5677: 5673: 5668: 5663: 5660: 5657: 5654: 5647: 5641: 5638: 5631: 5614: 5597: 5591: 5588: 5553: 5550: 5547: 5542: 5538: 5534: 5531: 5528: 5523: 5519: 5515: 5512: 5509: 5506: 5501: 5498: 5495: 5491: 5480: 5474: 5471: 5465: 5460: 5456: 5452: 5449: 5446: 5443: 5438: 5433: 5430: 5424: 5404: 5397: 5390: 5373: 5370: 5367: 5363: 5360: 5357: 5354: 5351: 5348: 5345: 5342: 5339: 5336: 5333: 5330: 5327: 5324: 5321: 5316: 5312: 5281: 5278: 5275: 5274: 5265: 5263: 5252: 5249: 5246: 5243: 5229: 5221: 5218: 5208: 5202: 5199: 5161: 5153: 5149: 5145: 5139: 5135: 5129: 5126: 5123: 5120: 5117: 5114: 5110: 5106: 5103: 5098: 5092: 5089: 5082: 5078: 5075: 5072: 5069: 5066: 5063: 5060: 5057: 5054: 5050: 5046: 5043: 5038: 5032: 5029: 5022: 5018: 5015: 5012: 4988: 4980: 4976: 4972: 4966: 4962: 4956: 4953: 4950: 4947: 4943: 4939: 4936: 4931: 4925: 4922: 4915: 4911: 4908: 4905: 4881: 4875: 4872: 4843: 4840: 4837: 4834: 4829: 4823: 4820: 4814: 4811: 4787: 4782: 4777: 4773: 4767: 4760: 4756: 4749: 4745: 4741: 4735: 4732: 4727: 4723: 4719: 4716: 4713: 4708: 4704: 4700: 4697: 4694: 4687: 4683: 4679: 4674: 4671: 4668: 4663: 4659: 4655: 4652: 4649: 4644: 4640: 4636: 4630: 4627: 4621: 4618: 4615: 4612: 4609: 4604: 4598: 4595: 4589: 4586: 4583: 4561: 4539: 4535: 4531: 4528: 4523: 4519: 4515: 4512: 4509: 4494: 4491: 4488: 4487: 4478: 4476: 4465: 4462: 4459: 4456: 4442: 4434: 4431: 4421: 4415: 4412: 4382: 4379: 4374: 4370: 4366: 4363: 4360: 4355: 4351: 4347: 4341: 4338: 4332: 4327: 4321: 4318: 4293: 4290: 4287: 4284: 4281: 4278: 4275: 4270: 4266: 4262: 4259: 4256: 4253: 4248: 4244: 4240: 4237: 4221: 4214: 4207: 4204: 4192: 4185: 4181: 4177: 4172: 4169: 4166: 4163: 4160: 4156: 4152: 4149: 4146: 4142: 4138: 4135: 4102:expected value 4070: 4067: 4064: 4061: 4056: as 4053: 4050: 4045: 4041: 4038: 4035: 4030: 4026: 3997: 3982:, named after 3976: 3973: 3954: 3945: 3941: 3935: 3931: 3928: 3925: 3922: 3919: 3916: 3913: 3910: 3907: 3904: 3901: 3898: 3895: 3892: 3887: 3883: 3879: 3876: 3871: 3866: 3863: 3860: 3856: 3850: 3847: 3841: 3835: 3832: 3829: 3825: 3807: 3806: 3795: 3792: 3789: 3786: 3774: 3771: 3768: 3765: 3762: 3758: 3754: 3751: 3748: 3745: 3742: 3739: 3735: 3711: 3678: 3640: 3625: 3579: 3576: 3574: 3573: 3487: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3460: 3456: 3434: 3431: 3428: 3425: 3422: 3419: 3416: 3412: 3408: 3386: 3383: 3380: 3377: 3374: 3371: 3368: 3364: 3360: 3357: 3354: 3326: 3323: 3320: 3317: 3314: 3311: 3308: 3305: 3301: 3297: 3283: 3252: 3249: 3246: 3243: 3240: 3237: 3234: 3230: 3226: 3210: 3204: 3190: 3185: 3182: 3179: 3176: 3173: 3171: 3165: 3162: 3159: 3156: 3153: 3150: 3147: 3144: 3141: 3137: 3132: 3129: 3127: 3124: 3121: 3118: 3115: 3114: 3111: 3108: 3105: 3102: 3100: 3094: 3091: 3088: 3085: 3082: 3079: 3076: 3072: 3067: 3064: 3062: 3059: 3056: 3053: 3050: 3047: 3044: 3043: 3041: 3024: 3013: 3010: 3007: 3004: 3001: 2998: 2995: 2990: 2987: 2983: 2977: 2971: 2967: 2963: 2960: 2957: 2952: 2948: 2939: 2934: 2931: 2928: 2924: 2917: 2913: 2908: 2902: 2898: 2894: 2891: 2888: 2883: 2879: 2872: 2868: 2842: 2839: 2835: 2829: 2825: 2821: 2818: 2815: 2810: 2806: 2790: 2777: 2774: 2769: 2766: 2763: 2758: 2755: 2751: 2745: 2739: 2735: 2731: 2728: 2725: 2722: 2719: 2711: 2706: 2703: 2700: 2696: 2689: 2685: 2680: 2674: 2670: 2666: 2663: 2660: 2657: 2654: 2648: 2644: 2635:, we can say: 2614: 2611: 2607: 2601: 2597: 2593: 2590: 2587: 2584: 2581: 2565: 2545: 2542: 2538: 2534: 2531: 2526: 2520: 2517: 2510: 2467: 2464: 2460: 2456: 2453: 2448: 2442: 2439: 2432: 2411: 2408: 2404: 2400: 2397: 2392: 2386: 2383: 2376: 2349: 2343: 2340: 2328:, the average 2317: 2314: 2291: 2288: 2285: 2282: 2277: 2273: 2269: 2266: 2263: 2256: 2252: 2248: 2241: 2236: 2233: 2230: 2226: 2209: 2201: 2200: 2191: 2189: 2178: 2175: 2164: 2154: 2147: 2141: 2138: 2130: 2125: 2122: 2119: 2114: 2108: 2105: 2044: 2041: 2037: 2033: 2030: 2025: 2019: 2016: 2008: 2005: 2002: 1998: 1993: 1988: 1973: 1972: 1963: 1961: 1950: 1947: 1944: 1941: 1927: 1916: 1906: 1900: 1897: 1865: 1862: 1846: 1843: 1840: 1836: 1832: 1812: 1809: 1806: 1802: 1796: 1792: 1767: 1764: 1761: 1758: 1755: 1752: 1749: 1745: 1741: 1738: 1691: 1688: 1684: 1679: 1676: 1672: 1668: 1665: 1660: 1654: 1651: 1644: 1638: 1633: 1628: 1625: 1622: 1618: 1598: 1597: 1588: 1586: 1575: 1572: 1569: 1566: 1552: 1544: 1541: 1531: 1525: 1522: 1481: 1480: 1471: 1470: 1462: 1461: 1453: 1452: 1451: 1450: 1449: 1447: 1444: 1400: 1395: 1390: 1386: 1380: 1373: 1369: 1362: 1358: 1354: 1348: 1345: 1340: 1336: 1332: 1329: 1326: 1321: 1317: 1313: 1310: 1307: 1300: 1296: 1292: 1287: 1284: 1281: 1276: 1272: 1268: 1265: 1262: 1257: 1253: 1249: 1243: 1240: 1234: 1231: 1228: 1225: 1222: 1217: 1211: 1208: 1202: 1199: 1196: 1174: 1152: 1148: 1144: 1141: 1136: 1132: 1128: 1125: 1122: 1091: 1084: 1077: 1076: 1067: 1065: 1054: 1051: 1048: 1045: 1031: 1028: 1023: 1017: 1014: 984: 979: 975: 971: 968: 965: 960: 956: 952: 947: 944: 939: 934: 928: 925: 904: 897: 883: 876: 851: 848: 839:expected value 763: 762: 751: 745: 738: 723: 720: 716:selection bias 668: 665: 525: 522: 517: 513: 510: 507: 504: 501: 498: 495: 492: 489: 486: 483: 467:expected value 458: 455: 346: 345: 343: 342: 335: 328: 320: 317: 316: 315: 314: 309: 301: 300: 299: 298: 293: 291:Bayes' theorem 288: 283: 278: 273: 265: 264: 263: 262: 257: 252: 247: 239: 238: 237: 236: 235: 234: 229: 224: 222:Observed value 219: 214: 209: 207:Expected value 204: 199: 189: 184: 183: 182: 177: 172: 167: 162: 157: 147: 146: 145: 135: 134: 133: 128: 123: 118: 113: 103: 98: 90: 89: 88: 87: 82: 77: 76: 75: 65: 64: 63: 50: 49: 41: 40: 34: 33: 22: 9: 6: 4: 3: 2: 10545: 10534: 10533:Large numbers 10531: 10529: 10526: 10524: 10521: 10519: 10516: 10514: 10511: 10510: 10508: 10499: 10498: 10492: 10489: 10487: 10483: 10479: 10476: 10471: 10470: 10465: 10462: 10457: 10452: 10451: 10446: 10443: 10438: 10434: 10430: 10429: 10424: 10420: 10419: 10408: 10404: 10400: 10396: 10391: 10386: 10382: 10378: 10374: 10370: 10366: 10361: 10357: 10351: 10347: 10342: 10338: 10334: 10329: 10325: 10320: 10316: 10314:87-91180-71-6 10310: 10306: 10302: 10297: 10293: 10288: 10284: 10282:0-19-853665-8 10278: 10274: 10269: 10268: 10250: 10244: 10240: 10236: 10232: 10228: 10221: 10219: 10210: 10203: 10195: 10191: 10186: 10181: 10177: 10173: 10169: 10162: 10154: 10150: 10146: 10142: 10138: 10134: 10130: 10123: 10114: 10109: 10105: 10101: 10097: 10090: 10083: 10078: 10067: 10060: 10046:on 2013-03-09 10042: 10035: 10028: 10014:on 2016-07-01 10010: 10006: 10002: 9995: 9988: 9980: 9978:9780387276052 9974: 9970: 9969: 9961: 9955: 9950: 9936: 9932: 9926: 9918: 9914: 9907: 9905: 9897: 9892: 9885: 9880: 9872: 9868: 9863: 9858: 9854: 9850: 9846: 9839: 9831: 9827: 9822: 9817: 9813: 9809: 9805: 9798: 9783: 9777: 9775: 9766: 9760: 9756: 9752: 9748: 9741: 9733: 9729: 9725: 9724:Yuri Prohorov 9719: 9717: 9709: 9704: 9702: 9693: 9687: 9679: 9675: 9671: 9667: 9663: 9660:(in French). 9659: 9655: 9648: 9640: 9636: 9632: 9628: 9624: 9620: 9613: 9604: 9600: 9593: 9586: 9578: 9571: 9563: 9556: 9548: 9544: 9540: 9536: 9531: 9526: 9522: 9518: 9514: 9507: 9499: 9497:9781852338961 9493: 9489: 9484: 9483: 9474: 9466: 9464:9781852338961 9460: 9456: 9451: 9450: 9441: 9433: 9427: 9423: 9416: 9408: 9404: 9400: 9396: 9392: 9388: 9381: 9370: 9363: 9361: 9359: 9357: 9355: 9353: 9351: 9349: 9347: 9338: 9334: 9330: 9326: 9322: 9318: 9314: 9310: 9303: 9295: 9293:9781852338961 9289: 9285: 9280: 9279: 9270: 9268: 9266: 9264: 9259: 9250: 9247: 9245: 9242: 9240: 9237: 9235: 9232: 9230: 9227: 9225: 9222: 9220: 9217: 9215: 9212: 9210: 9207: 9205: 9202: 9200: 9197: 9196: 9190: 9173: 9170: 9167: 9162: 9159: 9154: 9150: 9134: 9131: 9116: 9113: 9106: 9100: 9095: 9090: 9087: 9083: 9074: 9071: 9056: 9053: 9046: 9040: 9035: 9030: 9027: 9023: 9014: 9011: 8996: 8993: 8986: 8980: 8975: 8970: 8967: 8963: 8954: 8938: 8935: 8930: 8926: 8917: 8909: 8905: 8901: 8898: 8895: 8889: 8883: 8859: 8856: 8849: 8843: 8838: 8833: 8829: 8807: 8804: 8796: 8793: 8790: 8786: 8777: 8771: 8766: 8761: 8757: 8750: 8747: 8744: 8713: 8709: 8702: 8696: 8690: 8687: 8684: 8658: 8649: 8645: 8638: 8635: 8632: 8629: 8626: 8623: 8615: 8611: 8604: 8601: 8593: 8589: 8582: 8572: 8569: 8566: 8543: 8528: 8481: 8480: 8479: 8477: 8467: 8465: 8460: 8444: 8438: 8435: 8432: 8426: 8423: 8416: 8410: 8405: 8402: 8399: 8394: 8391: 8388: 8384: 8380: 8374: 8371: 8368: 8362: 8359: 8356: 8348: 8341: 8333: 8329: 8317: 8300: 8297: 8294: 8291: 8288: 8285: 8282: 8276: 8273: 8265: 8261: 8251: 8248: 8244: 8230: 8227: 8224: 8204: 8184: 8176: 8172: 8166: 8162: 8157: 8152: 8144: 8139: 8135: 8116: 8108: 8105: 8098: 8093: 8087: 8084: 8081: 8071: 8067: 8035: 8013: 8009: 8005: 8002: 7994: 7989: 7985: 7954: 7951: 7948: 7928: 7908: 7903: 7899: 7892: 7889: 7886: 7880: 7877: 7874: 7871: 7868: 7865: 7857: 7852: 7848: 7815: 7802: 7797: 7793: 7756: 7753: 7750: 7744: 7741: 7716: 7711: 7707: 7676: 7667: 7651: 7638: 7633: 7629: 7612: 7604: 7599: 7595: 7589: 7584: 7580: 7547: 7543: 7535:. Since the 7517: 7512: 7508: 7502: 7497: 7493: 7457: 7452: 7448: 7412: 7408: 7389: 7384: 7380: 7366: 7358: 7354: 7348: 7343: 7339: 7306: 7302: 7296: 7292: 7286: 7282: 7276: 7272: 7268: 7263: 7259: 7253: 7249: 7243: 7238: 7234: 7230: 7225: 7221: 7215: 7210: 7206: 7185: 7181: 7175: 7171: 7165: 7161: 7155: 7151: 7145: 7141: 7135: 7132: 7129: 7126: 7123: 7120: 7117: 7114: 7111: 7108: 7105: 7101: 7096: 7085: 7081: 7076: 7071: 7065: 7061: 7055: 7050: 7047: 7044: 7040: 7035: 7030: 7019: 7011: 7006: 7002: 6982: 6963: 6959: 6932: 6926: 6918: 6914: 6897: 6894: 6891: 6887: 6863: 6860: 6857: 6847: 6843: 6834: 6831: 6825: 6820: 6816: 6795: 6792: 6788: 6777: 6774: 6771: 6760: 6752: 6748: 6739: 6736: 6732: 6708: 6705: 6702: 6682: 6669: 6666: 6662: 6657: 6650: 6642: 6638: 6631: 6627: 6624: 6621: 6618: 6607: 6604: 6599: 6592: 6584: 6580: 6565: 6540: 6537: 6534: 6530: 6526: 6523: 6518: 6511: 6503: 6499: 6484: 6476: 6473: 6469: 6445: 6442: 6438: 6434: 6431: 6426: 6419: 6411: 6407: 6392: 6384: 6381: 6377: 6353: 6350: 6347: 6343: 6339: 6336: 6331: 6322: 6308: 6299: 6286: 6272: 6269: 6266: 6257: 6240: 6237: 6234: 6226: 6221: 6217: 6183: 6178: 6174: 6170: 6162: 6158: 6151: 6148: 6125: 6122: 6119: 6111: 6107: 6088: 6070: 6066: 6051: 6044: 6037: 6035: 6021: 6012: 5998: 5990: 5977: 5968: 5958: 5957: 5954: 5952: 5947: 5934: 5925: 5915: 5896: 5887: 5876: 5874: 5857: 5848: 5837: 5833: 5828: 5815: 5806: 5796: 5791: 5788: 5785: 5781: 5770: 5765: 5760: 5755: 5752: 5747: 5743: 5740: 5735: 5732: 5727: 5724: 5721: 5718: 5714: 5709: 5704: 5699: 5694: 5689: 5686: 5681: 5675: 5671: 5666: 5661: 5655: 5645: 5636: 5629: 5620: 5617: 5613: 5595: 5586: 5574: 5572: 5568: 5548: 5540: 5536: 5529: 5521: 5517: 5513: 5507: 5499: 5496: 5493: 5489: 5472: 5469: 5458: 5454: 5450: 5444: 5436: 5431: 5428: 5422: 5413: 5410: 5407: 5403: 5396: 5389: 5384: 5371: 5365: 5361: 5355: 5349: 5346: 5343: 5340: 5337: 5334: 5331: 5328: 5322: 5314: 5310: 5301: 5299: 5295: 5291: 5287: 5273: 5266: 5264: 5250: 5241: 5227: 5219: 5206: 5197: 5187: 5186: 5183: 5181: 5177: 5172: 5159: 5151: 5147: 5143: 5137: 5133: 5127: 5124: 5121: 5115: 5112: 5108: 5104: 5101: 5096: 5087: 5080: 5073: 5067: 5064: 5061: 5055: 5052: 5048: 5044: 5041: 5036: 5027: 5020: 5013: 5002: 4999: 4986: 4978: 4974: 4970: 4964: 4960: 4954: 4948: 4945: 4941: 4937: 4934: 4929: 4920: 4913: 4906: 4895: 4879: 4870: 4859: 4854: 4841: 4838: 4835: 4827: 4818: 4809: 4801: 4798: 4785: 4780: 4775: 4771: 4765: 4758: 4754: 4747: 4743: 4739: 4733: 4725: 4721: 4717: 4714: 4711: 4706: 4702: 4695: 4692: 4685: 4681: 4677: 4672: 4661: 4657: 4653: 4650: 4647: 4642: 4638: 4628: 4625: 4616: 4613: 4610: 4602: 4593: 4584: 4581: 4573: 4559: 4537: 4533: 4529: 4521: 4517: 4510: 4507: 4500: 4486: 4479: 4477: 4463: 4454: 4440: 4432: 4419: 4410: 4400: 4399: 4396: 4393: 4380: 4372: 4368: 4364: 4361: 4358: 4353: 4349: 4339: 4336: 4330: 4325: 4316: 4305: 4288: 4285: 4282: 4279: 4276: 4268: 4264: 4257: 4254: 4246: 4242: 4235: 4227: 4220: 4213: 4203: 4190: 4183: 4179: 4175: 4170: 4164: 4161: 4158: 4150: 4147: 4144: 4125: 4121: 4117: 4113: 4110: 4106: 4103: 4099: 4095: 4091: 4090: 4085: 4081: 4068: 4059: 4051: 4043: 4036: 4028: 4024: 4012: 4008: 4004: 4000: 3993: 3989: 3985: 3981: 3972: 3970: 3965: 3952: 3923: 3920: 3917: 3911: 3905: 3899: 3893: 3890: 3885: 3881: 3874: 3869: 3864: 3861: 3858: 3854: 3848: 3845: 3830: 3827: 3814: 3812: 3793: 3787: 3784: 3769: 3763: 3760: 3749: 3746: 3743: 3737: 3724: 3720: 3716: 3712: 3709: 3705: 3701: 3698: 3694: 3690: 3686: 3682: 3679: 3676: 3673: 3672: 3671: 3669: 3665: 3661: 3656: 3654: 3650: 3646: 3639: 3635: 3631: 3624: 3620: 3616: 3612: 3608: 3604: 3600: 3596: 3592: 3587: 3585: 3571: 3567: 3563: 3559: 3555: 3551: 3547: 3543: 3539: 3535: 3531: 3527: 3523: 3519: 3515: 3511: 3507: 3503: 3485: 3482: 3479: 3476: 3473: 3470: 3467: 3464: 3458: 3454: 3432: 3429: 3426: 3423: 3420: 3417: 3414: 3410: 3406: 3384: 3381: 3378: 3375: 3372: 3369: 3366: 3362: 3358: 3355: 3352: 3344: 3341: = 3340: 3324: 3321: 3318: 3315: 3312: 3309: 3306: 3303: 3299: 3295: 3286: 3282: 3268: 3250: 3247: 3244: 3241: 3238: 3235: 3232: 3228: 3224: 3213: 3209: 3205: 3183: 3180: 3177: 3174: 3169: 3160: 3157: 3151: 3148: 3145: 3142: 3139: 3135: 3130: 3122: 3116: 3109: 3106: 3103: 3098: 3089: 3083: 3080: 3077: 3074: 3070: 3065: 3057: 3051: 3048: 3045: 3039: 3029: 3025: 3008: 3002: 2999: 2996: 2993: 2988: 2985: 2981: 2975: 2969: 2961: 2958: 2950: 2946: 2932: 2929: 2926: 2922: 2915: 2911: 2906: 2900: 2892: 2889: 2881: 2877: 2870: 2866: 2858: 2840: 2837: 2833: 2827: 2819: 2816: 2808: 2804: 2795: 2794:geometrically 2791: 2775: 2772: 2767: 2764: 2761: 2756: 2753: 2749: 2743: 2737: 2733: 2726: 2720: 2717: 2704: 2701: 2698: 2694: 2687: 2683: 2678: 2672: 2668: 2661: 2655: 2652: 2646: 2642: 2634: 2630: 2612: 2609: 2605: 2599: 2595: 2588: 2582: 2579: 2571: 2570:exponentially 2567: 2566: 2564: 2561: 2559: 2543: 2540: 2532: 2529: 2524: 2515: 2497: 2492: 2491:almost surely 2488: 2483: 2481: 2465: 2462: 2454: 2451: 2446: 2437: 2409: 2406: 2398: 2395: 2390: 2381: 2365: 2347: 2338: 2327: 2323: 2313: 2311: 2307: 2302: 2289: 2283: 2275: 2271: 2264: 2261: 2254: 2250: 2246: 2234: 2231: 2228: 2224: 2215: 2212: 2208: 2199: 2192: 2190: 2176: 2173: 2145: 2136: 2123: 2117: 2112: 2103: 2093: 2092: 2089: 2086: 2084: 2078: 2076: 2070: 2068: 2063: 2060: 2055: 2042: 2039: 2035: 2031: 2028: 2023: 2014: 2000: 1991: 1978: 1971: 1964: 1962: 1948: 1939: 1925: 1904: 1895: 1885: 1884: 1881: 1879: 1875: 1872:(also called 1871: 1861: 1858: 1844: 1841: 1838: 1834: 1830: 1810: 1807: 1804: 1800: 1794: 1790: 1781: 1762: 1759: 1756: 1750: 1747: 1743: 1739: 1736: 1728: 1724: 1720: 1717:of the first 1715: 1709: 1707: 1702: 1689: 1686: 1682: 1677: 1674: 1666: 1663: 1658: 1649: 1636: 1620: 1607: 1605: 1596: 1589: 1587: 1573: 1564: 1550: 1542: 1529: 1520: 1510: 1509: 1506: 1504: 1500: 1496: 1493:(also called 1492: 1475: 1466: 1457: 1443: 1441: 1436: 1434: 1430: 1426: 1422: 1420: 1419:not necessary 1416: 1411: 1398: 1393: 1388: 1384: 1378: 1371: 1367: 1360: 1356: 1352: 1346: 1338: 1334: 1330: 1327: 1324: 1319: 1315: 1308: 1305: 1298: 1294: 1290: 1285: 1274: 1270: 1266: 1263: 1260: 1255: 1251: 1241: 1238: 1229: 1226: 1223: 1215: 1206: 1197: 1194: 1186: 1172: 1150: 1146: 1142: 1134: 1130: 1123: 1120: 1113: 1108: 1106: 1102: 1098: 1094: 1087: 1075: 1068: 1066: 1052: 1043: 1029: 1021: 1012: 1002: 1001: 998: 995: 977: 973: 969: 966: 963: 958: 954: 945: 942: 937: 932: 923: 912: 910: 903: 896: 892: 889: 882: 875: 871: 869: 864: 862: 857: 847: 845: 840: 836: 832: 828: 824: 820: 816: 811: 808: 803: 802:S. D. Poisson 799: 795: 791: 787: 782: 781: 775: 771: 760: 755: 752: 749: 746: 743: 740: 739: 736: 732: 728: 719: 717: 712: 710: 706: 702: 698: 694: 690: 686: 682: 678: 674: 664: 661: 657: 654: 653:computational 650: 645: 642: 627: 625: 611: 608: 607:almost surely 604: 580: 570: 566: 564: 560: 556: 552: 548: 543: 541: 536: 523: 520: 515: 511: 508: 505: 502: 499: 496: 493: 490: 487: 484: 481: 470: 468: 464: 454: 451: 449: 445: 441: 437: 433: 428: 426: 421: 416: 412: 408: 404: 401: 396: 394: 390: 385: 381: 377: 373: 369: 361: 357: 352: 341: 336: 334: 329: 327: 322: 321: 319: 318: 313: 310: 308: 305: 304: 303: 302: 297: 294: 292: 289: 287: 284: 282: 279: 277: 274: 272: 269: 268: 267: 266: 261: 258: 256: 253: 251: 248: 246: 243: 242: 241: 240: 233: 230: 228: 225: 223: 220: 218: 215: 213: 210: 208: 205: 203: 200: 198: 195: 194: 193: 190: 188: 185: 181: 178: 176: 173: 171: 168: 166: 163: 161: 158: 156: 153: 152: 151: 148: 144: 141: 140: 139: 136: 132: 129: 127: 124: 122: 119: 117: 114: 112: 109: 108: 107: 104: 102: 99: 97: 94: 93: 92: 91: 86: 83: 81: 80:Indeterminism 78: 74: 71: 70: 69: 66: 62: 59: 58: 57: 54: 53: 52: 51: 47: 43: 42: 39: 36: 35: 32: 28: 27: 19: 10494: 10467: 10448: 10426: 10380: 10376: 10364: 10345: 10336: 10323: 10304: 10300: 10291: 10272: 10252:, retrieved 10230: 10208: 10202: 10175: 10171: 10161: 10136: 10132: 10122: 10103: 10099: 10089: 10077: 10059: 10048:. Retrieved 10041:the original 10027: 10016:. Retrieved 10009:the original 10004: 10000: 9987: 9971:. Springer. 9967: 9960: 9949: 9938:. Retrieved 9934: 9925: 9916: 9891: 9879: 9852: 9848: 9838: 9811: 9807: 9797: 9786:. Retrieved 9746: 9740: 9734:. EMS Press. 9731: 9686:cite journal 9661: 9657: 9647: 9622: 9618: 9612: 9598: 9585: 9576: 9570: 9561: 9555: 9520: 9516: 9506: 9481: 9473: 9448: 9440: 9421: 9415: 9390: 9386: 9380: 9312: 9308: 9302: 9277: 9234:Lindy effect 9135: 9132: 9075: 9072: 9015: 9012: 8955: 8875: 8529:Evaluate f(X 8510:= a+(b - a)U 8473: 8470:Applications 8461: 8318: 8257: 8254:Consequences 8249: 8245: 7668: 6984:We compute 6983: 6287: 6258: 6057: 6049: 6038: 5948: 5877: 5831: 5829: 5621: 5615: 5611: 5610:in terms of 5575: 5570: 5566: 5414: 5411: 5405: 5401: 5394: 5387: 5385: 5302: 5297: 5283: 5267: 5175: 5173: 5003: 5000: 4896: 4855: 4802: 4799: 4574: 4496: 4480: 4394: 4306: 4218: 4211: 4209: 4126: 4119: 4111: 4104: 4100:with finite 4093: 4087: 4086: 4082: 4010: 4006: 4002: 3995: 3991: 3987: 3979: 3978: 3966: 3815: 3810: 3808: 3722: 3718: 3707: 3703: 3699: 3692: 3688: 3684: 3680: 3674: 3667: 3663: 3659: 3657: 3652: 3648: 3644: 3637: 3633: 3629: 3622: 3618: 3614: 3610: 3606: 3605:defined for 3598: 3594: 3590: 3588: 3583: 3581: 3569: 3565: 3561: 3557: 3553: 3549: 3545: 3541: 3537: 3533: 3529: 3525: 3521: 3517: 3513: 3509: 3505: 3501: 3342: 3338: 3284: 3280: 3266: 3211: 3207: 2568:Let X be an 2562: 2557: 2495: 2486: 2484: 2479: 2363: 2325: 2321: 2319: 2305: 2303: 2216: 2210: 2206: 2204: 2193: 2087: 2082: 2079: 2071: 2064: 2058: 2056: 1979: 1976: 1965: 1869: 1867: 1859: 1722: 1718: 1710: 1705: 1703: 1608: 1603: 1601: 1590: 1490: 1488: 1437: 1423: 1418: 1412: 1187: 1109: 1096: 1089: 1082: 1080: 1069: 996: 913: 908: 901: 894: 880: 873: 867: 866: 860: 859: 855: 853: 812: 789: 785: 767: 753: 747: 741: 713: 708: 704: 692: 684: 672: 670: 646: 628: 623: 602: 576: 558: 544: 537: 471: 460: 452: 447: 443: 439: 435: 431: 429: 420:large number 419: 397: 375: 371: 365: 356:illustration 312:Tree diagram 307:Venn diagram 285: 271:Independence 217:Markov chain 101:Sample space 10084:, Lemma 2.4 9954:Ross (2009) 9935:builtin.com 9708:Seneta 2013 9130:= 1.000194 8552:), ..., f(X 8537:), ..., f(X 4894:results in 4116:real number 3984:Émile Borel 3677:is compact, 2792:Let X be a 2308:, see e.g. 844:convergence 800:. In 1837, 759:Fick's laws 689:heavy tails 605:flips will 540:sample mean 463:probability 450:increases. 389:sample mean 227:Random walk 68:Determinism 56:Probability 10507:Categories 10495:explained 10264:References 10254:2023-12-08 10050:2014-06-28 10018:2014-06-28 9940:2023-10-20 9896:Loève 1977 9884:Loève 1977 9788:2012-06-09 7967:such that 7669:There are 6721:, we have 6553:Note that 5830:The limit 3715:dominating 3697:almost all 3601:) is some 2487:strong law 1874:Kolmogorov 1864:Strong law 1780:asymptotic 907:) = ... = 861:strong law 831:Kolmogorov 667:Limitation 656:algorithms 138:Experiment 85:Randomness 31:statistics 10486:animation 10469:MathWorld 10450:MathWorld 10433:EMS Press 10390:1309.6488 10377:Bernoulli 10194:122166046 9871:0091-1798 9830:122166046 9678:120850863 9539:0003-4851 9329:1063-6706 9244:Sortition 9171:− 9160:− 9088:− 9084:∫ 9028:− 9024:∫ 8968:− 8964:∫ 8830:∫ 8794:− 8758:∫ 8748:− 8688:− 8570:− 8464:histogram 8430:≈ 8392:− 8385:∫ 8372:∈ 8354:≈ 8286:− 8225:ϵ 8163:ϵ 8153:≤ 8109:ϵ 8094:≤ 8088:ϵ 8082:≥ 8003:≤ 7900:σ 7890:− 7872:τ 7830:, and so 7754:− 7133:≤ 7109:≤ 7102:∑ 7041:∑ 6930:∞ 6903:∞ 6888:∑ 6864:ϵ 6858:≥ 6832:ω 6778:ϵ 6772:≥ 6761:ω 6737:ω 6703:ϵ 6670:ϵ 6667:≥ 6651:ω 6619:ϵ 6616:∃ 6612:⟺ 6605:≠ 6593:ω 6572:∞ 6569:→ 6524:≠ 6512:ω 6491:∞ 6488:→ 6474:ω 6420:ω 6399:∞ 6396:→ 6382:ω 6326:¯ 6315:∞ 6312:→ 6267:μ 6244:∞ 6238:τ 6187:∞ 6175:σ 6152:⁡ 6129:∞ 6123:μ 6019:∞ 6016:→ 5999:μ 5988:→ 5972:¯ 5932:∞ 5929:→ 5916:μ 5905:→ 5891:¯ 5852:¯ 5813:∞ 5810:→ 5792:μ 5777:→ 5728:μ 5672:φ 5640:¯ 5630:φ 5590:¯ 5537:φ 5518:φ 5490:φ 5455:φ 5423:φ 5369:→ 5344:μ 5311:φ 5248:∞ 5245:→ 5228:μ 5217:→ 5201:¯ 5148:ε 5134:σ 5128:− 5122:≥ 5116:ε 5113:≥ 5105:μ 5102:− 5091:¯ 5074:⁡ 5068:− 5056:ε 5045:μ 5042:− 5031:¯ 5014:⁡ 4975:ε 4961:σ 4955:≤ 4949:ε 4946:≥ 4938:μ 4935:− 4924:¯ 4907:⁡ 4874:¯ 4839:μ 4822:¯ 4772:σ 4744:σ 4715:⋯ 4696:⁡ 4651:⋯ 4617:⁡ 4597:¯ 4585:⁡ 4552:(for all 4534:σ 4511:⁡ 4461:∞ 4458:→ 4441:μ 4430:→ 4414:¯ 4362:⋯ 4320:¯ 4292:∞ 4286:μ 4280:⋯ 4171:≤ 4165:σ 4159:≥ 4151:μ 4148:− 4066:∞ 4063:→ 4049:→ 3940:→ 3924:θ 3906:⁡ 3900:− 3894:θ 3855:∑ 3834:Θ 3831:∈ 3828:θ 3791:Θ 3788:∈ 3785:θ 3761:≤ 3750:θ 3717:function 3664:uniformly 3649:pointwise 3483:⁡ 3477:⁡ 3471:⁡ 3430:⁡ 3424:⁡ 3418:⁡ 3382:⁡ 3376:⁡ 3370:⁡ 3356:⁡ 3319:⁡ 3313:⁡ 3307:⁡ 3248:⁡ 3242:⁡ 3236:⁡ 3181:− 3178:≤ 3158:− 3152:⁡ 3140:− 3107:≥ 3084:⁡ 3049:− 3003:⁡ 2997:− 2986:− 2959:− 2938:∞ 2923:∑ 2890:− 2838:− 2817:− 2773:π 2754:− 2721:⁡ 2710:∞ 2695:∫ 2656:⁡ 2610:− 2583:⁡ 2544:ε 2533:μ 2530:− 2519:¯ 2463:≠ 2455:μ 2452:− 2441:¯ 2410:ε 2399:μ 2396:− 2385:¯ 2342:¯ 2287:∞ 2265:⁡ 2240:∞ 2225:∑ 2163:⟶ 2140:¯ 2124:⁡ 2118:− 2107:¯ 2083:something 2032:μ 2018:¯ 2007:∞ 2004:→ 1977:That is, 1946:∞ 1943:→ 1926:μ 1915:⟶ 1899:¯ 1842:⁡ 1808:⁡ 1751:⁡ 1714:Chebyshev 1712:shown by 1678:ε 1667:μ 1664:− 1653:¯ 1627:∞ 1624:→ 1571:∞ 1568:→ 1551:μ 1540:→ 1524:¯ 1385:σ 1357:σ 1328:⋯ 1309:⁡ 1264:⋯ 1230:⁡ 1210:¯ 1198:⁡ 1165:(for all 1147:σ 1124:⁡ 1050:∞ 1047:→ 1030:μ 1027:→ 1016:¯ 967:⋯ 927:¯ 815:Chebyshev 731:Diffusion 579:fair coin 409:may lose 131:Singleton 10484:package 10407:88520834 9407:18521840 9193:See also 8522:, ..., X 8502:, ..., U 8490:, ..., X 4499:variance 4109:variance 3934:‖ 3840:‖ 3757:‖ 3734:‖ 3706:at each 3695:∈ Θ for 3603:function 3589:Suppose 3288:is then 2478:for all 2322:weak law 1495:Khinchin 1446:Weak law 1415:variance 1112:variance 868:weak law 865:and the 835:Khinchin 827:Cantelli 679:or some 610:converge 457:Examples 415:roulette 212:Variance 10435:, 2001 10153:2323947 9639:2709176 9547:2239008 9337:2238905 3778:for all 3274:⁄ 3026:If the 754:Bottom: 748:Middle: 722:History 697:tangent 634:⁄ 617:⁄ 596:⁄ 586:⁄ 432:average 384:average 378:) is a 126:Outcome 10405: 10352: 10311: 10279: 10245: 10192: 10151: 9975: 9869: 9828: 9761: 9676: 9637: 9595:) in: 9545: 9537: 9494: 9461: 9428: 9405: 9335: 9327: 9290: 9286:–190. 9138:f(x) = 8548:), f(X 8533:), f(X 6673: 6199:, and 6010: 5996: 5983: 5875:to μ: 5292:, the 5239: 5225: 5212: 4856:Using 4452: 4438: 4425: 4226:i.i.d. 4210:Given 4122:> 0 4092:. Let 3950: 3813:, and 3782: 2919: 2857:series 2691: 2498:> 0 2171: 2158: 1937: 1923: 1910: 1562: 1548: 1535: 1041: 900:) = E( 819:Markov 735:solute 701:median 407:casino 403:events 400:random 370:, the 73:System 61:Axioms 10403:S2CID 10385:arXiv 10303:[ 10190:S2CID 10149:JSTOR 10069:(PDF) 10044:(PDF) 10037:(PDF) 10012:(PDF) 9997:(PDF) 9855:(2). 9826:S2CID 9674:S2CID 9635:JSTOR 9543:JSTOR 9403:S2CID 9372:(PDF) 9333:S2CID 9255:Notes 4096:be a 3670:. If 850:Forms 823:Borel 411:money 106:Event 10350:ISBN 10309:ISBN 10277:ISBN 10243:ISBN 9973:ISBN 9867:ISSN 9759:ISBN 9692:link 9662:1846 9535:ISSN 9492:ISBN 9459:ISBN 9426:ISBN 9325:ISSN 9288:ISBN 9013:and 8228:> 8197:for 8028:for 7952:> 7734:and 7666:. 7475:and 6927:< 6706:> 6622:> 6241:< 6184:< 6126:< 6085:are 6005:when 5569:and 5386:All 5288:for 5234:when 5053:< 4447:when 4289:< 3651:(in 3206:Let 2541:< 2485:The 2407:> 2320:The 2284:< 2166:a.s. 2073:See 1932:when 1918:a.s. 1868:The 1675:< 1557:when 1489:The 833:and 742:Top: 393:mean 10395:doi 10235:doi 10180:doi 10141:doi 10108:doi 9857:doi 9816:doi 9751:doi 9666:doi 9627:doi 9525:doi 9395:doi 9317:doi 9284:181 8518:, X 8498:, U 8486:, X 6562:lim 6481:lim 6389:lim 6366:or 6305:lim 6149:Var 6087:iid 5921:for 5565:if 5484:and 5284:By 5174:As 4860:on 4693:Var 4614:Var 4582:Var 4508:Var 3971:). 3824:sup 3666:in 3632:), 3480:log 3474:log 3468:log 3427:log 3421:log 3415:log 3379:log 3373:log 3367:log 3353:log 3316:log 3310:log 3304:log 3245:log 3239:log 3233:log 2718:sin 2653:sin 2580:sin 2482:). 2262:Var 1997:lim 1839:log 1805:log 1782:to 1748:log 1617:lim 1431:or 1417:is 1306:Var 1227:Var 1195:Var 1121:Var 1107:.) 1097:not 622:as 612:to 524:3.5 446:as 438:of 436:sum 427:). 395:. 376:LLN 366:In 360:die 354:An 10509:: 10466:. 10447:. 10431:, 10425:, 10401:. 10393:. 10381:19 10379:. 10241:, 10229:, 10217:^ 10188:. 10176:55 10174:. 10170:. 10147:. 10137:98 10135:. 10131:. 10104:40 10102:. 10098:. 10005:13 10003:. 9999:. 9933:. 9915:. 9903:^ 9865:. 9851:. 9847:. 9824:. 9812:55 9810:. 9806:. 9773:^ 9757:. 9730:. 9726:. 9715:^ 9700:^ 9688:}} 9684:{{ 9656:. 9633:. 9623:44 9621:. 9541:. 9533:. 9521:38 9519:. 9515:. 9490:. 9488:63 9457:. 9455:92 9401:. 9389:. 9345:^ 9331:. 9323:. 9313:24 9311:. 9262:^ 8734:= 8466:. 8056:Pr 6981:. 6953:Pr 6908:Pr 6796:0. 6729:Pr 6466:Pr 6446:1. 6374:Pr 6295:Pr 6256:. 6235:=: 6141:, 6120:=: 6089:, 5838:, 5802:as 5619:: 5409:. 5393:, 5372:0. 4217:, 4134:Pr 4124:, 3953:0. 3586:. 3149:ln 3081:ln 3000:ln 2077:. 2043:1. 1987:Pr 1690:1. 1632:Pr 1606:, 1442:. 1036:as 879:, 829:, 825:, 821:, 817:, 10482:R 10472:. 10453:. 10409:. 10397:: 10387:: 10358:. 10317:. 10285:. 10237:: 10211:. 10196:. 10182:: 10155:. 10143:: 10116:. 10110:: 10071:. 10053:. 10021:. 9981:. 9943:. 9919:. 9873:. 9859:: 9853:6 9832:. 9818:: 9791:. 9767:. 9753:: 9710:. 9694:) 9680:. 9668:: 9641:. 9629:: 9605:. 9603:7 9549:. 9527:: 9500:. 9467:. 9434:. 9409:. 9397:: 9391:6 9374:. 9339:. 9319:: 9296:. 9174:1 9168:e 9163:1 9155:x 9151:e 9117:x 9114:d 9110:) 9107:x 9104:( 9101:f 9096:2 9091:1 9057:x 9054:d 9050:) 9047:x 9044:( 9041:f 9036:2 9031:1 8997:x 8994:d 8990:) 8987:x 8984:( 8981:f 8976:2 8971:1 8939:1 8936:+ 8931:3 8927:x 8921:) 8918:x 8915:( 8910:2 8906:s 8902:o 8899:c 8896:= 8893:) 8890:x 8887:( 8884:f 8860:x 8857:d 8853:) 8850:x 8847:( 8844:f 8839:b 8834:a 8821:= 8808:x 8805:d 8797:a 8791:b 8787:1 8781:) 8778:x 8775:( 8772:f 8767:b 8762:a 8754:) 8751:a 8745:b 8742:( 8722:) 8719:) 8714:1 8710:X 8706:( 8703:f 8700:( 8697:E 8694:) 8691:a 8685:b 8682:( 8659:n 8655:) 8650:n 8646:X 8642:( 8639:f 8636:+ 8633:. 8630:. 8627:. 8624:+ 8621:) 8616:2 8612:X 8608:( 8605:f 8602:+ 8599:) 8594:1 8590:X 8586:( 8583:f 8576:) 8573:a 8567:b 8564:( 8554:n 8550:2 8546:1 8541:) 8539:n 8535:2 8531:1 8524:n 8520:2 8516:1 8512:i 8508:i 8504:n 8500:2 8496:1 8492:n 8488:2 8484:1 8448:) 8445:a 8442:( 8439:f 8436:h 8433:2 8427:x 8424:d 8420:) 8417:x 8414:( 8411:f 8406:h 8403:+ 8400:a 8395:h 8389:a 8381:= 8378:) 8375:C 8369:X 8366:( 8363:P 8360:= 8357:p 8349:n 8345:) 8342:C 8339:( 8334:n 8330:N 8304:] 8301:h 8298:+ 8295:a 8292:, 8289:h 8283:a 8280:( 8277:= 8274:C 8231:0 8205:n 8185:, 8177:2 8173:n 8167:4 8158:C 8150:] 8145:4 8140:n 8136:S 8132:[ 8127:E 8117:4 8113:) 8106:n 8103:( 8099:1 8091:) 8085:n 8078:| 8072:n 8068:S 8063:| 8059:( 8036:n 8014:2 8010:n 8006:C 8000:] 7995:4 7990:n 7986:S 7982:[ 7977:E 7955:0 7949:C 7929:n 7909:. 7904:4 7896:) 7893:1 7887:n 7884:( 7881:n 7878:3 7875:+ 7869:n 7866:= 7863:] 7858:4 7853:n 7849:S 7845:[ 7840:E 7816:2 7812:) 7808:] 7803:2 7798:i 7794:X 7790:[ 7785:E 7780:( 7760:) 7757:1 7751:n 7748:( 7745:n 7742:3 7722:] 7717:4 7712:i 7708:X 7704:[ 7699:E 7677:n 7652:2 7648:) 7644:] 7639:2 7634:i 7630:X 7626:[ 7621:E 7616:( 7613:= 7610:] 7605:2 7600:j 7596:X 7590:2 7585:i 7581:X 7577:[ 7572:E 7548:i 7544:X 7523:] 7518:2 7513:j 7509:X 7503:2 7498:i 7494:X 7490:[ 7485:E 7463:] 7458:4 7453:i 7449:X 7445:[ 7440:E 7418:] 7413:j 7409:X 7405:[ 7400:E 7395:] 7390:3 7385:i 7381:X 7377:[ 7372:E 7367:= 7364:] 7359:j 7355:X 7349:3 7344:i 7340:X 7336:[ 7331:E 7307:l 7303:X 7297:k 7293:X 7287:j 7283:X 7277:i 7273:X 7269:, 7264:k 7260:X 7254:j 7250:X 7244:2 7239:i 7235:X 7231:, 7226:j 7222:X 7216:3 7211:i 7207:X 7186:. 7182:] 7176:l 7172:X 7166:k 7162:X 7156:j 7152:X 7146:i 7142:X 7136:n 7130:l 7127:, 7124:k 7121:, 7118:j 7115:, 7112:i 7106:1 7097:[ 7091:E 7086:= 7082:] 7077:4 7072:) 7066:i 7062:X 7056:n 7051:1 7048:= 7045:i 7036:( 7031:[ 7025:E 7020:= 7017:] 7012:4 7007:n 7003:S 6999:[ 6994:E 6969:) 6964:n 6960:A 6956:( 6933:, 6924:) 6919:n 6915:A 6911:( 6898:1 6895:= 6892:n 6867:} 6861:n 6854:| 6848:n 6844:S 6839:| 6835:: 6829:{ 6826:= 6821:n 6817:A 6793:= 6789:) 6775:n 6768:| 6764:) 6758:( 6753:n 6749:S 6744:| 6740:: 6733:( 6709:0 6683:, 6663:| 6658:n 6654:) 6648:( 6643:n 6639:S 6632:| 6628:, 6625:0 6608:0 6600:n 6596:) 6590:( 6585:n 6581:S 6566:n 6541:, 6538:0 6535:= 6531:) 6527:0 6519:n 6515:) 6509:( 6504:n 6500:S 6485:n 6477:: 6470:( 6443:= 6439:) 6435:0 6432:= 6427:n 6423:) 6417:( 6412:n 6408:S 6393:n 6385:: 6378:( 6354:, 6351:1 6348:= 6344:) 6340:0 6337:= 6332:n 6323:X 6309:n 6300:( 6273:0 6270:= 6232:] 6227:4 6222:i 6218:X 6214:[ 6209:E 6179:2 6171:= 6168:) 6163:i 6159:X 6155:( 6117:] 6112:i 6108:X 6104:[ 6099:E 6071:i 6067:X 6043:) 6041:2 6039:( 6022:. 6013:n 5991:P 5978:n 5969:X 5935:. 5926:n 5909:D 5897:n 5888:X 5858:n 5849:X 5832:e 5816:. 5807:n 5797:, 5789:t 5786:i 5782:e 5771:n 5766:] 5761:) 5756:n 5753:t 5748:( 5744:o 5741:+ 5736:n 5733:t 5725:i 5722:+ 5719:1 5715:[ 5710:= 5705:n 5700:] 5695:) 5690:n 5687:t 5682:( 5676:X 5667:[ 5662:= 5659:) 5656:t 5653:( 5646:n 5637:X 5616:X 5612:φ 5596:n 5587:X 5571:Y 5567:X 5552:) 5549:t 5546:( 5541:Y 5533:) 5530:t 5527:( 5522:X 5514:= 5511:) 5508:t 5505:( 5500:Y 5497:+ 5494:X 5479:) 5473:n 5470:t 5464:( 5459:X 5451:= 5448:) 5445:t 5442:( 5437:X 5432:n 5429:1 5406:X 5402:φ 5398:2 5395:X 5391:1 5388:X 5366:t 5362:, 5359:) 5356:t 5353:( 5350:o 5347:+ 5341:t 5338:i 5335:+ 5332:1 5329:= 5326:) 5323:t 5320:( 5315:X 5298:X 5272:) 5270:2 5268:( 5251:. 5242:n 5220:P 5207:n 5198:X 5176:n 5160:. 5152:2 5144:n 5138:2 5125:1 5119:) 5109:| 5097:n 5088:X 5081:| 5077:( 5071:P 5065:1 5062:= 5059:) 5049:| 5037:n 5028:X 5021:| 5017:( 5011:P 4987:. 4979:2 4971:n 4965:2 4952:) 4942:| 4930:n 4921:X 4914:| 4910:( 4904:P 4880:n 4871:X 4842:. 4836:= 4833:) 4828:n 4819:X 4813:( 4810:E 4786:. 4781:n 4776:2 4766:= 4759:2 4755:n 4748:2 4740:n 4734:= 4731:) 4726:n 4722:X 4718:+ 4712:+ 4707:1 4703:X 4699:( 4686:2 4682:n 4678:1 4673:= 4670:) 4667:) 4662:n 4658:X 4654:+ 4648:+ 4643:1 4639:X 4635:( 4629:n 4626:1 4620:( 4611:= 4608:) 4603:n 4594:X 4588:( 4560:i 4538:2 4530:= 4527:) 4522:i 4518:X 4514:( 4485:) 4483:2 4481:( 4464:. 4455:n 4433:P 4420:n 4411:X 4381:. 4378:) 4373:n 4369:X 4365:+ 4359:+ 4354:1 4350:X 4346:( 4340:n 4337:1 4331:= 4326:n 4317:X 4283:= 4277:= 4274:) 4269:2 4265:X 4261:( 4258:E 4255:= 4252:) 4247:1 4243:X 4239:( 4236:E 4222:2 4219:X 4215:1 4212:X 4191:. 4184:2 4180:k 4176:1 4168:) 4162:k 4155:| 4145:X 4141:| 4137:( 4120:k 4112:σ 4105:μ 4094:X 4069:. 4060:n 4052:p 4044:n 4040:) 4037:E 4034:( 4029:n 4025:N 4011:n 4007:E 4003:E 4001:( 3998:n 3996:N 3992:p 3988:E 3944:P 3930:] 3927:) 3921:, 3918:X 3915:( 3912:f 3909:[ 3903:E 3897:) 3891:, 3886:i 3882:X 3878:( 3875:f 3870:n 3865:1 3862:= 3859:i 3849:n 3846:1 3811:θ 3794:. 3773:) 3770:x 3767:( 3764:d 3753:) 3747:, 3744:x 3741:( 3738:f 3723:x 3721:( 3719:d 3710:. 3708:θ 3704:x 3700:x 3693:θ 3689:θ 3687:, 3685:x 3683:( 3681:f 3675:Θ 3668:θ 3653:θ 3645:θ 3643:, 3641:2 3638:X 3636:( 3634:f 3630:θ 3628:, 3626:1 3623:X 3621:( 3619:f 3615:θ 3611:θ 3607:θ 3599:θ 3597:, 3595:x 3593:( 3591:f 3570:ε 3568:( 3566:p 3562:ε 3560:( 3558:p 3554:m 3550:n 3546:m 3542:ε 3540:( 3538:p 3534:n 3530:ε 3528:( 3526:p 3522:ε 3520:( 3518:p 3514:ε 3510:n 3506:n 3502:ε 3486:n 3465:2 3459:/ 3455:1 3433:k 3411:/ 3407:k 3385:n 3363:/ 3359:n 3343:n 3339:k 3325:. 3322:k 3300:/ 3296:k 3285:k 3281:X 3276:2 3272:1 3267:k 3251:k 3229:/ 3225:k 3212:k 3208:X 3184:e 3175:x 3170:, 3164:) 3161:x 3155:( 3146:x 3143:2 3136:e 3131:= 3126:) 3123:x 3120:( 3117:F 3110:e 3104:x 3099:, 3093:) 3090:x 3087:( 3078:x 3075:2 3071:e 3066:= 3061:) 3058:x 3055:( 3052:F 3046:1 3040:{ 3012:) 3009:2 3006:( 2994:= 2989:x 2982:2 2976:x 2970:x 2966:) 2962:1 2956:( 2951:x 2947:2 2933:1 2930:= 2927:x 2916:= 2912:) 2907:X 2901:X 2897:) 2893:1 2887:( 2882:X 2878:2 2871:( 2867:E 2841:1 2834:X 2828:X 2824:) 2820:1 2814:( 2809:X 2805:2 2776:2 2768:= 2765:x 2762:d 2757:x 2750:e 2744:x 2738:x 2734:e 2730:) 2727:x 2724:( 2705:0 2702:= 2699:x 2688:= 2684:) 2679:X 2673:X 2669:e 2665:) 2662:X 2659:( 2647:( 2643:E 2613:1 2606:X 2600:X 2596:e 2592:) 2589:X 2586:( 2558:n 2537:| 2525:n 2516:X 2509:| 2496:ε 2480:n 2466:0 2459:| 2447:n 2438:X 2431:| 2403:| 2391:n 2382:X 2375:| 2364:μ 2348:n 2339:X 2326:n 2290:. 2281:] 2276:k 2272:X 2268:[ 2255:2 2251:k 2247:1 2235:1 2232:= 2229:k 2211:k 2207:X 2198:) 2196:2 2194:( 2177:, 2174:0 2153:] 2146:n 2137:X 2129:[ 2121:E 2113:n 2104:X 2059:n 2040:= 2036:) 2029:= 2024:n 2015:X 2001:n 1992:( 1970:) 1968:3 1966:( 1949:. 1940:n 1905:n 1896:X 1845:n 1835:/ 1831:1 1811:n 1801:/ 1795:2 1791:n 1766:) 1763:1 1760:+ 1757:n 1754:( 1744:/ 1740:n 1737:2 1723:n 1719:n 1706:ε 1687:= 1683:) 1671:| 1659:n 1650:X 1643:| 1637:( 1621:n 1604:ε 1595:) 1593:2 1591:( 1574:. 1565:n 1543:P 1530:n 1521:X 1399:. 1394:n 1389:2 1379:= 1372:2 1368:n 1361:2 1353:n 1347:= 1344:) 1339:n 1335:X 1331:+ 1325:+ 1320:1 1316:X 1312:( 1299:2 1295:n 1291:1 1286:= 1283:) 1280:) 1275:n 1271:X 1267:+ 1261:+ 1256:1 1252:X 1248:( 1242:n 1239:1 1233:( 1224:= 1221:) 1216:n 1207:X 1201:( 1173:i 1151:2 1143:= 1140:) 1135:i 1131:X 1127:( 1092:j 1090:X 1085:j 1083:X 1074:) 1072:1 1070:( 1053:. 1044:n 1022:n 1013:X 983:) 978:n 974:X 970:+ 964:+ 959:1 955:X 951:( 946:n 943:1 938:= 933:n 924:X 909:μ 905:2 902:X 898:1 895:X 884:2 881:X 877:1 874:X 784:( 709:n 705:n 693:α 685:n 673:n 636:2 632:1 624:n 619:2 615:1 603:n 598:2 594:1 588:2 584:1 559:n 521:= 516:6 512:6 509:+ 506:5 503:+ 500:4 497:+ 494:3 491:+ 488:2 485:+ 482:1 448:n 444:n 440:n 374:( 339:e 332:t 325:v 20:.

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Knowledge

Law of large numbers

Index