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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 standardised difference converges in distribution to a standard normal random variable.
The weak law of large numbers states that if X1, X2, X3, ... is an infinite sequence of random variables, all of which have the same expected value μ and the same finite varianceThis article is about mathematics. Alternate meaning: variance (land use). In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically σ2, and they are uncorrelatedIn probability theory and statistics, to call two real-valued random variables X and Y uncorrelated means that their correlation is zero, or, equivalently, their covariance is zero. If X and Y are independent then they are uncorrelated. It is not true, ho (i.e., the correlationIn probability theory and statistics, the correlation also called correlation coefficient between two random variables is found by dividing their covariance by the product of their standard deviations. It is defined only if these standard deviations are f between any two of them is zero), then the sample average
Somewhat less tersely: For any positive number ε, no matter how small, we have
A consequence of the weak law of large numbers is the asymptotic equipartition propertyInformation theory The asymptotic equipartition property (AEP or Shannon-McMillan theorem, is a direct consequence of the weak law of large numbers and is used extensively in information theory. It can be summed up by the phrase 'Almost everything is almo.
The strong law of large numbers states that if X1, X2, X3, ... is an infinite sequence of random variables that are independent and identically distributed with common expected value μ, and if E(|X1|) < ∞, then
i.e., the sample average converges almost surely to μ.
If we replace the finite expectation condition with a finite second moment condition, E(X12) < ∞, then we obtain both almost sure convergence and convergence in mean square. In either case, these conditions also imply the consequent of the weak law of large numbers, since almost sure convergence implies convergence in probability (as, indeed, does convergence in mean square).
This law justifies the intuitive interpretation of the expected value of a random variable as the "long-term average when sampling repeatedly".
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