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Probability distributions cheat sheet

The CS1 distributions on one page: what each one models, its density, mean, variance and generating function, all properly rendered. If you can say when to reach for each of these and quote its first two moments, a surprising share of the paper is already yours.

Discrete distributions

Counts of things: claims, deaths, successes. The generating function quoted is the PGF, E[z^X].

Bernoulli

A single yes/no trial: one policy claims or it does not.

pmf

P(X=x) = px (1-p)1-x

PGF

1 - p + pz

Mean

Variance

p(1-p)

Binomial

n, p

Number of successes in n independent trials with the same p.

pmf

P(X=x) = n!x!(n-x)! px (1-p)n-x

PGF

(1 - p + pz)n

Mean

Variance

np(1-p)

A sum of n independent Bernoulli(p) variables.

Geometric

Trials up to and including the first success (support 1, 2, 3, ...).

pmf

P(X=x) = p(1-p)x-1

PGF

pz1-(1-p)z

Mean

Variance

1-pp2

Memoryless: the only discrete distribution that forgets how long you have waited.

Negative binomial

k, p

Trials up to and including the k-th success; also the classic overdispersed claim count.

pmf

P(X=x) = (x-1)!(k-1)!(x-k)! pk (1-p)x-k

PGF

(pz1-(1-p)z)k

Mean

Variance

k(1-p)p2

A sum of k independent geometrics. Variance exceeds the mean, unlike the Poisson.

Poisson

Counts of rare events in a fixed exposure: the default claim-number model.

pmf

P(X=x) = λx e-λx!

PGF

eλ(z-1)

Mean

Variance

Mean equals variance. Sums of independent Poissons are Poisson.

Continuous distributions

Amounts and waiting times: claim severities, lifetimes, estimators. The generating function quoted is the MGF, E[e^(tX)], where it exists.

Uniform

a, b

Complete ignorance between two bounds; the building block of simulation.

pdf

f(x) = 1b-a

Mean

a+b2

Variance

(b-a)212

Exponential

Waiting time to the first event of a Poisson process.

pdf

f(x) = λ e-λ x

MGF

λλ - t

Mean

1λ

Variance

1λ2

Memoryless, and a constant force of mortality gives exponential lifetimes.

Gamma

α, λ

Sums of exponential waiting times; a standard claim severity model.

pdf

f(x) = λαΓ(α) xα - 1 e-λ x

MGF

(λλ - t)α

Mean

αλ

Variance

αλ2

Gamma(1, λ) is exponential; Gamma(k/2, 1/2) is chi-squared with k degrees of freedom.

Normal

μ, σ2

The central limit theorem's destination: sums, averages, estimators.

pdf

f(x) = 1σ √(2pi) e-(x-μ)22sigma2

MGF

eμ t + σ2 t22

Mean

Variance

σ2

Lognormal

μ, σ2

X = e^Z with Z normal: share prices and heavy-ish claim severities.

pdf

f(x) = 1x σ √(2pi) e-(ln x-μ)22sigma2

Mean

eμ + σ22

Variance

e2mu + σ2(eσ2 - 1)

No MGF exists. The parameters are the mean and variance of ln X, not of X.

Beta

α, β

Random proportions on (0, 1); the conjugate prior for a binomial p in credibility work.

pdf

f(x) = Γ(α+β)Γ(α)Γ(β) xα-1(1-x)β-1

Mean

αα+β

Variance

α β(α+β)2 (α+β+1)

Sampling distributions

Built from normals, met in estimation and hypothesis testing rather than as models of data. Interactive tables and calculators for all three live on our statistical tables page.

Chi-squared

Sum of k squared standard normals; variance estimates and goodness-of-fit tests.

pdf

χ2k = Z21 + ... + Z2k

Mean

Variance

A Gamma(k/2, 1/2) distribution.

Student's t

A normal mean estimated with an unknown variance: small-sample tests and intervals.

pdf

t_ν = Z√(χ2ν / ν)

Mean

Variance

νν - 2

Heavier tails than the normal; variance exists only for ν > 2. Approaches N(0,1) as ν grows.

F

d₁, d₂

Ratio of two variance estimates: comparing variances and ANOVA.

pdf

F = (χ2d₁)/(d₁)(χ2d₂)/(d₂)

Mean

d₂d₂ - 2

Variance

2d22 (d₁ + d₂ - 2)d₁ (d₂-2)2 (d₂-4)

Mean exists for d₂ > 2, variance for d₂ > 4. 1/F swaps the degrees of freedom.

Want these as flashcards instead of a list? This page is exactly the kind of content Memori was built for: every formula here can be typed as a plain-text Memori formula card, and the shop carries a ready-made CS1 set. Join the beta, or work the sampling distributions properly with our statistical tables.