A2. Bernoulli process and related distributions

A Bernoulli process is characterized by a probability $p$ of success, to which is associated the uncertain number $X=1$, and probability $1-p$ of failure, to which is associated the uncertain number $X=0$. Therefore, technically, a Bernoulli distribution is just a binomial with $n=1$. But conceptually it is very important, because it is the basic process from which other distributions arise:

• a binomial distribution describes the probability of the total number of successes in $n$ independent Bernoulli trials `having' (or more precisely `believed to have') the same probability of success $p$;
• a geometric distribution describes the probability (again assuming independence and constant $p$) of the trial at which³⁶ the first success occurs;
• a Pascal distribution (or negative binomial distribution) concerns finally the trial at which the $k$-th success occurs.³⁷