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Distribution Cheat Sheet

Every distribution worth knowing, in one comparable grid: what it models, when to reach for it, its PMF or PDF, and its mean and variance. This is a lookup companion to the probability track, intuition first and formulas as scaffolding. The real memory trick is not eighteen islands but one family tree.

Rapid Recall

Discrete distributions carry probability in a mass function (height is probability); continuous ones carry it in a density (area is probability, a single point has probability zero). Almost everything sprouts from Bernoulli trials and the Normal: sum Bernoullis to get a binomial, drop replacement for the hypergeometric, push trials high and probability low for the Poisson. Geometric and exponential are the memoryless twins, negative binomial and gamma are their "wait for the rth event" generalizations, and squared normals build the chi-squared, which feeds Student's t and the F. The central limit theorem is the grand unifier that bends sums of many small effects toward the Normal.

What this section covers

  • Discrete Distributions: the eight integer-outcome distributions, from Bernoulli to the multinomial.
  • Continuous Distributions: the ten real-valued distributions, from the uniform to the multivariate normal.
  • Big Ideas: the threads that compress the whole grid into a few reusable rules.

How they all connect

This map is the real memory trick. Do not memorize 18 islands, memorize one family tree. Almost everything sprouts from Bernoulli trials and the Normal.

  • Sum of \(n\) independent Bernoulli(p) trials equals Binomial(n,p).
  • Binomial done without replacement becomes Hypergeometric.
  • Binomial with \(n\) huge, \(p\) tiny, \(np=\lambda\) fixed limits to Poisson(\(\lambda\)).
  • Geometric is wait for the 1st success; Negative Binomial is wait for the \(r\)-th success (Geometric is NB with \(r=1\)).
  • Geometric is the discrete memoryless distribution; Exponential is the continuous one, same idea, different clock.
  • Exponential is wait for the 1st event; Gamma is wait for the \(\alpha\)-th event (a sum of \(\alpha\) exponentials). Exponential is Gamma with \(\alpha=1\).
  • Sum of \(k\) squared standard Normals equals Chi-Squared(\(k\)), which is itself a special Gamma.
  • A standard Normal divided by \(\sqrt{\chi^2/\nu}\) equals Student's t(\(\nu\)); as \(\nu\to\infty\), t goes to Normal.
  • Ratio of two standard Normals equals Cauchy equals Student's t with \(\nu=1\).
  • Multinomial generalizes Binomial to \(k\) categories; Multivariate Normal generalizes Normal to \(d\) correlated dimensions.
  • Beta is the conjugate prior for Binomial; Gamma is the conjugate prior for Poisson. (Bayesian pairings.)
  • The grand unifier, the Central Limit Theorem: the average of many independent things, whatever their distribution, drifts toward the Normal. (Cauchy is the famous exception, because it has no mean.)
flowchart TB
  Bern["Bernoulli(p)"] --> Bin["Binomial(n,p)"]
  Bin -->|no replacement| HG["Hypergeometric"]
  Bin -->|"n→∞, p→0, np=λ"| Pois["Poisson(λ)"]
  Bern --> Geom["Geometric"]
  Geom -->|wait for r| NB["Negative Binomial"]
  Geom -. discrete↔continuous .-> Exp["Exponential"]
  Exp -->|wait for α| Gamma["Gamma"]
  Norm["Normal"] -->|sum of k squares| Chi["Chi-Squared(k)"]
  Chi --> tdist["Student's t"]
  Norm --> tdist
  Norm -->|ratio of two| Cauchy["Cauchy = t(1)"]
  Gamma --> Chi
  Norm -->|d correlated dims| MVN["Multivariate Normal"]
  CLT["Central Limit Theorem"] --> Norm