Relationships among probability distributions

In probability theory and statistics, there are several relationships among probability distributions. These relations can be categorized in the following groups:

One distribution is a special case of another with a broader parameter space Transforms (function of a random variable); Combinations (function of several variables); Approximation (limit) relationships; Compound relationships (useful for Bayesian inference); Duality; Conjugate priors.

Special case of distribution parametrization A binomial distribution with parameters n = 1 and p is a Bernoulli distribution with parameter p. A negative binomial distribution with parameters n = 1 and p is a geometric distribution with parameter p. A gamma distribution with shape parameter α = 1 and rate parameter β is an exponential distribution with rate parameter β. A gamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom. A chi-squared distribution with 2 degrees of freedom (k = 2) is an exponential distribution with a mean value of 2 (rate λ = 1/2 .) A Weibull distribution with shape parameter k = 1 and rate parameter β is an exponential distribution with rate parameter β. A beta distribution with shape parameters α = β = 1 is a continuous uniform distribution over the real numbers 0 to 1. A beta-binomial distribution with parameter n and shape parameters α = β = 1 is a discrete uniform distribution over the integers 0 to n. A Student's t-distribution with one degree of freedom (v = 1) is a Cauchy distribution with location parameter x = 0 and scale parameter γ = 1. A Burr distribution with parameters c = 1 and k (and scale λ) is a Lomax distribution with shape k (and scale λ.)

Transform of a variable

Multiple of a random variable Multiplying the variable by any positive real constant yields a scaling of the original distribution. Some are self-replicating, meaning that the scaling yields the same family of distributions, albeit with a different parameter: normal distribution, gamma distribution, Cauchy distribution, exponential distribution, Erlang distribution, Weibull distribution, logistic distribution, error distribution, power-law distribution, Rayleigh distribution. Example:

If X is a gamma random variable with shape and rate parameters (α, β), then Y = aX is a gamma random variable with parameters (α,β/a). If X is a gamma random variable with shape and scale parameters (α, θ), then Y = aX is a gamma random variable with parameters (α,aθ).

Linear function of a random variable The affine transform ax + b yields a relocation and scaling of the original distribution. The following are self-replicating: Normal distribution, Cauchy distribution, Logistic distribution, Error distribution, Power distribution, Rayleigh distribution. Example:

If Z is a normal random variable with parameters (μ = m, σ2 = s2), then X = aZ + b is a normal random variable with parameters (μ = am + b, σ2 = a2s2).

Reciprocal of a random variable The reciprocal 1/X of a random variable X, is a member of the same family of distribution as X, in the following cases: Cauchy distribution, F distribution, log logistic distribution. Examples:

If X is a Cauchy (μ, σ) random variable, then 1/X is a Cauchy (μ/C, σ/C) random variable where C = μ2 + σ2. If X is an F(ν1, ν2) random variable then 1/X is an F(ν2, ν1) random variable.

Other cases Some distributions are invariant under a specific transformation. Example:

If X is a beta (α, β) random variable then (1 − X) is a beta (β, α) random variable. If X is a binomial (n, p) random variable then (n − X) is a binomial (n, 1 − p) random variable. If X follows a continuous uniform distribution on [0,1], and FX(X) is its cumulative distribution function (CDF), then the random variable U=FX(X) follows a standard uniform distribution on [0,1]. Some distributions are variant under a specific transformation.

If X is a normal (μ, σ2) random variable then eX is a lognormal (μ, σ2) random variable. Conversely, if X is a lognormal (μ, σ2) random variable then log X is a normal (μ, σ2) random variable. If X is an exponential random variable with mean β, then X1/γ is a Weibull (γ, β) random variable. The square of a standard normal random variable has a chi-squared distribution with one degree of freedom. If X is a Student’s t random variable with ν degree of freedom, then X2 is an F (1,ν) random variable. If X is a double exponential random variable with mean 0 and scale λ, then |X| is an exponential random variable with mean λ. A geometric random variable is the floor of an exponential random variable. A rectangular random variable is the floor of a uniform random variable. A reciprocal random variable is the exponential of a uniform random variable.

Functions of several variables

Sum of variables

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