### Statistical Mechanics Lecture 14 of 29

April 27, 2012 by Markus Mueller

M. Mueller, ICTP

This lesson was devoted to the study of the distribution of functions of n random variables xi, i=1,...,n. A crucial concept is that of an estimate or estimator. An estimator is a random variable, function of the xi, whose expectation value gives an approximation to a parameter, usually unknown, that characterizes the statistical system.

The average A of the random variables xi is an estimate for the expectation value of each of the xi if they are equally distributed. It was shown that N times the expectation value of A is equal to the sum of the expectation values of the xi. This result does not depend on whether the variables are correlated or not, that is, the first moment tells us nothing about correlation of random variables.

For independent variables, the variance V of the sum S of the xi, S = n*A, is equal to the sum of the variances. If the xi are equally distributed with variance v then V = nv. The variance of A is equal to v/n in this case. This implies that as n -> ?, the variance, and therefore the standard deviation, of A tends to zero.

If one doesn't know the distribution of the xi, still assumed equally distributed and independent, then an estimate for the variance is required. That is, another random variable whose expectation value yields the variance of the distribution. One such random variable was devised that doesn't coincide with the naive guess.

The Law of Large Numbers was introduced.

This is the central limit theorem

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