Probability

Lesson 6 of 14

Common Discrete Distributions

Bernoulli, binomial, geometric, and Poisson: when each applies and their main formulas.

Bernoulli and binomial

Bernoulli(p): a single trial, 1 with probability pp, else 0. E[X]=pE[X] = p, Var(X)=p(1−p)\text{Var}(X) = p(1-p).

Binomial(n, p): number of successes in nn independent trials.

P(X=k)=(nk)pk(1−p)n−k,E[X]=np,Var(X)=np(1−p)P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad E[X] = np, \quad \text{Var}(X) = np(1-p)

The moments follow instantly from linearity: a binomial is a sum of nn independent Bernoullis.