M. Mueller, ICTP

This lesson was devoted to the study of the distribution of functions of *n* random variables *x _{i}, i=1,...,n*. A crucial concept is that of an

**estimate or estimator**. An estimator is a random variable, function of the

*x*, whose expectation value gives an approximation to a parameter, usually unknown, that characterizes the statistical system.

_{i}The average *A *of the random variables *x _{i}* is an estimate for the expectation value of each of the

*x*if they are equally distributed. It was shown that

_{i}*N*times the expectation value of

*A*is equal to the sum of the expectation values of the

*x*. 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.

_{i}For independent variables, the **variance** *V* of the sum *S* of the *x _{i}, S = *

*n*A*, is equal to the sum of the variances. If the

*x*are equally distributed with variance

_{i}*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 *x _{i}*, 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**