Statistical Mechanics Lecture 12 of 29

April 23, 2012 by Marco Bardoscia

M. Bardoscia, ICTP

This lesson started with the probabilistic analysis of the problem of tossing a coin, a conventional example in probability theory. The probability of getting k heads after n trials was computed assuming that p is the probability that the coin will come out heads in each trial (1/2 for a fair coin). It was argued that the average number of heads after n trials is np and then shown through direct computation. As a matter of fact, for the case of the binomial distribution, all the moments can be computed. This is not the case for every distribution. In order for all the moments to be computable, the distribution must vanish faster than any polinomial.

Continuous random variables were introduced in the second part of the lesson. The analogue of probability distributions for continuous variables are probability density functions. Moments were explored for the normal, constant and Lorentz(Cauchy) probability densities.

Three numbers a, b and c are chosen randomly from the real positive line. What is the probability that one can form a triangle with sides a, b and c?

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