| Law Of Large Numbers |
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In Probability Theory , several laws of large numbers say that the average of a Sequence of Random Variables with a common Distribution Converges (in the senses given below) to their common Expectation , in the Limit as the size of the sequence goes to infinity. Various formulations of the law of large numbers, and their associated conditions, specify convergence in different ways. When the random variables have a finite Variance , the Central Limit Theorem extends our understanding of the convergence of their average by describing the distribution of the standardised difference between the sum of the random variables and the expectation of this sum. Regardless of the underlying distribution of the random variables, this standardized difference Converges In Distribution to a Standard Normal Random Variable . The phrase "law of large numbers" is also sometimes used to refer to the principle that the probability of any possible event (even an unlikely one) occurring ''at least once'' in a series increases with the number of events in the series. For example, the odds that you will win the lottery are very low; however, the odds that ''someone'' will win the lottery are quite good, provided that a large enough number of people purchased lottery tickets. THE WEAK LAW The weak law of large numbers states that if ''X''1, ''X''2, ''X''3, ... is an infinite Sequence of Random Variable s, where all the random variables have the same Expected Value μ and Variance σ2; and are Uncorrelated (i.e., the Correlation between any two of them is zero), then the sample average : Converges In Probability to μ. Somewhat less tersely: For any positive number ε, no matter how small, we have | ||
| \operatorname{P}( \left \overline{X} N-\mu Ight \leq Arepsilon) | 1 - \operatorname{P}( \left \overline{X}_n-\mu
ight > arepsilon) \geq 1 - \operatorname{P}( \left \overline{X}_n-\mu
ight \geq arepsilon) \geq 1 - rac{\sigma^2}{ arepsilon^2 n} |
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| The '''strong Law Of Large Numbers''' States That If ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, Is An Infinite Sequence Of Random Variables That Are | "http://wwwinformationdelightinfo/encyclopedia/entry/Independent_(probability)" class="copylinks">Independent and identically distributed with E(''X''<sub>i</sub>) < ∞   (and where the common expected value is μ), then |
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