Common probability distributions

Most random variables that show up in practice belong to one of a handful of named families, each pinned down by just one or two parameters instead of an arbitrary probability table. The binomial distribution counts successes across n independent yes/no trials, each with the same success probability p — its mean np and variance np(1−p) follow directly from adding up n independent Bernoulli trials one at a time. The Poisson distribution is what a binomial count turns into when n grows large and p shrinks so that np stays fixed: a count of rare events (radioactive decays, typos per page) over a continuum, fixed by a single rate parameter.

The normal distribution earns its ubiquity from the central limit theorem: whenever a measured quantity is really the sum of many small independent effects, its distribution is approximately bell-shaped regardless of what the individual effects look like — which is a reason to use it, not a licence to assume it applies to literally everything. When the sample is small and the population's standard deviation must itself be estimated from the data, Student's t distribution — heavier-tailed than the normal, converging to it as sample size grows — is the honest choice instead.

Knowing which family fits is the first step in almost any statistical calculation; getting the variance formula right (np(1−p), not np) is the detail that trips people up most often, because it is genuinely smaller than the mean for every p strictly between 0 and 1, and equal to it only in the limiting, non-random case p = 0 or p = 1.