### Statistical Mechanics Lecture 15 of 29

June 25, 2012 by Matteo Marsili

M. Marsili, ICTP

This lesson starts showing how to use the random variable theory to estimates errors when you perform an experiment by calculating the variance of the random variable X and the variance of its mean value <X>. Then the characteristics functions or generating functions were introduced as the Fourier transform of the probability density function. These characteristics functions are very useful because they can be used to find moments of a random variable. Provided that the nth moment exists, it can be calculated by differentiating n times the characteristic function.

Two examples were considered the Binomial distribution and the Normal distribution, for this two cases the characteristic function was calculated. Then It was obtained the first moment of the Binomial distribution and it was demonstrated that for the Normal distribution all the odd moments are zero while e generic formula can be obtain for the even moments.

In the following some properties of the characteristic function were presented and they were applied to calculate the characteristic function of a sum of random variables in two cases the Normal distribution and the Poisson distribution. it was shown that the sum of normal distributed variables is also a normal variable and the sum of Poisson distributed variables is also a Poisson variable.

It was introduced the cumulants ?n of a probability distribution, a set of quantities that provide an alternative to the moments of the distribution. The relation between cumulants and moments for the first moments were obtained and it was shown that for the Normal distribution just the first two cumulants are different from zero. Finally the proof of the Central limit theorem was given.

Calculate the characteristic function and the first three moments of the Uniform distribution.

Using the characteristic function formalism calculate that the second moment for the Binomial distribution.

Calculate the first three cumulants of the Uniform and the Poisson distributions.

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