Diagram of relationships between probability distributions

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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.

More mathematical diagrams

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.

Parameterizations

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 σ2 are 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. Mean and variance are undefined for the Cauchy distribution.

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.

Relationships

In all statements about two random variables, the random variables are implicitly independent .

Geometric / negative binomial : If each Xi is geometric random variable with probability of success p then the sum of n Xi‘s is a negative binomial random variable with parameters n and p.

Negative binomial / geometric : A negative binomial distribution with r = 1 is a geometric distribution.

Negative binomial / Poisson : If X has a negative binomial random variable with r large, p near 1, and r(1 − p) = λ, then FX ≈ FY where Y is a Poisson random variable with mean λ.

Beta-binomial / discrete uniform : A beta-binomial (n, 1, 1) random variable is a discrete uniform random variable over the values 0 … n.

Beta-binomial / binomial : Let X be a beta-binomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large. If Y is a binomial(n, p) random variable then FX ≈ FY.

Hypergeometric / binomial : The difference between a hypergeometric distribution and a binomial distribution is the difference between sampling without replacement and sampling with replacement. As the population size increases relative to the sample size, the difference becomes negligible.

Geometric / geometric : If X1 and X2 are geometric random variables with probability of success p1 and p2 respectively, then min(X1, X2) is a geometric random variable with probability of success p = p1 + p2 − p1 p2. The relationship is simpler in terms of failure probabilities: q = q1 q2.

Poisson / Poisson : If X1 and X2 are Poisson random variables with means μ1 and μ2 respectively, then X1 + X2 is a Poisson random variable with mean μ1 + μ2.

Binomial / Poisson : If X is a binomial(n, p) random variable and Y is a Poisson(np) distribution then P(X = n) ≈ P(Y = n) if n is large and np is small. For more information, see Poisson approximation to binomial.

Binomial / Bernoulli : If X is a binomial(n, p) random...