Every distribution is a shape, and a story about how it arose.
Two ingredients recur everywhere in probability: counts of discrete events, and measurements on a continuum. From a small set of mechanisms, dozens of named distributions follow. This guide collects 24 of them, lets you bend their parameters, traces how each descends from the others, and lets you draw from them by hand.
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I
The atlas
probability mass · density · cumulative
Each plate shows one distribution. Move its parameters and the curve answers; the moments beneath are computed in closed form, not estimated.
II
The lines of descent
limits · sums · transforms · conjugacy
The named distributions are not separate inventions. They connect: one becomes another in a limit, as a sum, under a transform, or as a conjugate prior. Follow an edge to read the relation.
limitsum or counttransformconjugacy
Bernoulli → Binomial · sum / count
Sum of n independent Bernoulli(p) trials is Binomial(n, p). The binomial counts the successes.
III
Drawing by hand
samples · convergence · the central limit
A distribution is a recipe for randomness. Draw from one and the histogram fills toward its curve. Average the draws in groups and a different law appears, regardless of where you started.