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*?