Editing Law of large numbers (section)

===Weak law===
{{multiple image |width1=50 |image1=Blank300.png
|width2=100 |image2=Lawoflargenumbersanimation2.gif |footer=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 approaches 50% as the number of trials increases.
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The '''weak law of large numbers''' (also called [[Aleksandr Khinchin|Khinchin]]'s law) states that given a collection of [[Independent and identically distributed random variables|independent and identically distributed]] (iid) samples from a random variable with finite mean, the sample mean [[Convergence in probability|converges in probability]] to the expected value<ref>{{harvnb|Loève|1977|loc=Chapter 1.4, p. 14}}</ref>
{{NumBlk||<math display="block">
    \overline{X}_n\ \overset{P}{\rightarrow}\ \mu \qquad\textrm{when}\ n \to \infty.
</math>|{{EquationRef|2}}}}

That is, for any positive number ''ε'',

<math display="block">
    \lim_{n\to\infty}\Pr\!\left(\,|\overline{X}_n-\mu| < \varepsilon\,\right) = 1.
</math>

Interpreting this result, the weak law states that for any nonzero margin specified (''ε''), 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.

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 shown by [[Pafnuty Chebyshev|Chebyshev]] 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 of the first ''n'' values goes to zero as ''n'' goes to infinity.<ref name=EncMath/> As an example, assume that each random variable in the series follows a [[Gaussian distribution]] (normal distribution) with mean zero, but with variance equal to <math>2n/\log(n+1)</math>, 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 [[asymptotic]] to <math>n^2 / \log n</math>. The variance of the average is therefore asymptotic to <math>1 / \log n</math> and goes to zero.

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