Probability distributions have a surprising number inter-connections. A dashed line in the chart below indicates an approximate (limit) relationship between two distribution families. A solid line indicates an exact relationship: special case, sum, or transformation.

Click on a distribution for the parameterization of that distribution. Click on an arrow for details on the relationship represented by the arrow. Other diagrams on this site:

The chart above is adapted from the chart originally published by Lawrence Leemis in 1986 (Relationships Among Common Univariate Distributions, American Statistician 40:143-146.) Leemis published a larger chart in 2008 which is available online.

The precise relationships between distributions depend on parameterization. The relationships detailed below depend on the following parameterizations for the PDFs.

Let C(n, k) denote the binomial coefficient(n, k) and B(a, b) = Γ(a) Γ(b) / Γ(a + b).

Geometric: f(x) = p (1-p)x for non-negative integers x.

Discrete uniform: f(x) = 1/n for x = 1, 2, ..., n.

Negative binomial: f(x) = C(r + x - 1, x) pr(1-p)x for non-negative integers x. See notes on the negative binomial distribution.

Beta binomial: f(x) = C(n, x) B(α + x, n + β - x) / B(α, β) for x = 0, 1, ..., n.

Hypergeometric: f(x) = C(M, x) C(N-M, K - x) / C(N, K) for x = 0, 1, ..., N.

Poisson: f(x) = exp(-λ) λx/ x! for non-negative integers x. The parameter λ is both the mean and the variance.

Binomial: f(x) = C(n, x) px(1 - p)n-x for x = 0, 1, ..., n.

Bernoulli: f(x) = px(1 - p)1-x where x = 0 or 1.

Lognormal: f(x) = (2πσ2)-1/2 exp( -(log(x) - μ)2/ 2σ2) / x for positive x. Note that μ and σ2are not the mean and variance of the distribution.

Normal : f(x) = (2π σ2)-1/2 exp( - ½((x - μ)/σ)2 ) for all x.

Beta: f(x) = Γ(α + β) xα-1(1 - x)β-1 / (Γ(α) Γ(β)) for 0 ≤ x ≤ 1.

Standard normal: f(x) = (2π)-1/2 exp( -x2/2) for all x.

Chi-squared: f(x) = x-ν/2-1 exp(-x/2) / Γ(ν/2) 2ν/2 for positive x. The parameter ν is called the degrees of freedom.

Gamma: f(x) = β-α xα-1 exp(-x/β) / Γ(α) for positive x. The parameter α is called the shape and β is the scale.

Uniform: f(x) = 1 for 0 ≤ x ≤ 1.

Cauchy: f(x) = σ/(π( (x - μ)2 + σ2) ) for all x. Note that μ and σ are location and scale parameters. The Cauchy distribution has no mean or variance.

Snedecor F: f(x) is proportional to x(ν1 - 2)/2 / (1 + (ν1/ν2) x)(ν1 + ν2)/2 for positive x.

Exponential: f(x) = exp(-x/μ)/μ for positive x. The parameter μ is the mean.

Student t: f(x) is proportional to (1 + (x2/ν))-(ν + 1)/2 for positive x. The parameter ν is called the degrees of freedom.

Weibull: f(x) = (γ/β) xγ-1 exp(- xγ/β) for positive x. The parameter γ is the shape and β is the scale.

Double exponential : f(x) = exp(-|x-μ|/σ) / 2σ for all x. The parameter μ is the location and mean; σ is the scale.

For comparison, see distribution parameterizations in R/S-PLUS and Mathematica.