Bernoulli and Binomial

One-shot trials and the count of successes across many.

A Bernoulli $(p)$ random variable is the simplest interesting random variable: it equals $1$ with probability $p$ and $0$ otherwise. Mean and variance are

E[X] = p, \qquad \mathrm{Var}(X) = p(1 - p)

Variance is maximized at $p = 0.5$ , where outcomes are most uncertain.

A Binomial $(n, p)$ random variable counts the number of successes in $n$ independent Bernoulli $(p)$ trials. Its PMF is

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \dots, n

Mean and variance scale by $n$ :

E[X] = np, \qquad \mathrm{Var}(X) = np(1 - p)

The Binomial is the building block for an enormous range of discrete models: A/B test conversion counts, defaults in a credit portfolio, up-moves in a binomial option-pricing tree. Two convenient closure properties: sums of independent Bernoullis with the same $p$ are Binomial, and sums of independent Binomials sharing $p$ are also Binomial.