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.
Contents
Special case of distribution parametrization
 A binomial (n, p) random variable with n = 1, is a Bernoulli (p) random variable.
 A negative binomial distribution with r = 1 is a geometric distribution.
 A gamma distribution with shape parameter α = 1 and scale parameter β is an exponential (β) distribution.
 A gamma (α, β) random variable with α = ν/2 and β = 2, is a chisquared random variable with ν degrees of freedom.
 A chisquared distribution with 2 degrees of freedom is an exponential distribution with mean 2 and vice versa.
 A Weibull (1, β) random variable is an exponential random variable with mean β.
 A beta random variable with parameters α = β = 1 is a uniform random variable.
 A betabinomial (n, 1, 1) random variable is a discrete uniform random variable over the values 0 ... n.
 A random variable with a t distribution with one degree of freedom is a Cauchy(0,1) random variable.
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 selfreplicating, 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 distribution, Rayleigh distribution.
Example:
 If X is a gamma random variable with parameters (r, λ), then Y=aX is a gamma random variable with parameters (r, aλ).
Linear function of a random variable
The affine transfom ax + b yields a relocation and scaling of the original distribution. The following are selfreplicating: Normal distribution, Cauchy distribution, Logistic distribution, Error distribution, Power distribution, Rayleigh distribution.
Example:
 If Z is a normal random variable with parameters (μ=m, σ^{2}=s^{2}), then X=aZ+b is a normal random variable with parameters (μ=am+b, σ^{2}=a^{2}s^{2}).
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 = μ^{2} + σ^{2}.
 If X is an F(ν_{1}, ν_{2}) random variable then 1/X is an F(ν_{2}, ν_{1}) 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, 1p) random variable.
 If X has cumulative distribution function F_{X}, then F_{X}(X) is a standard uniform (0,1) random variable
 If X is a normal (μ, σ^{2}) random variable then e^{X} is a lognormal (μ, σ^{2}) random variable.
 Conversely, if X is a lognormal (μ, σ^{2}) random variable then log X is a normal (μ, σ^{2}) random variable.
 If X is an exponential random variable with mean β, then X^{1/γ} is a Weibull (γ, β) random variable.
 The square of a standard normal random variable has a chisquared distribution with one degree of freedom.
 If X is a Student’s t random variable with ν degree of freedom, then X^{2} 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.
Functions of several variables
Sum of variables
The distribution of the sum of independent random variables is called the convolution of the primal distribution.
 If it has a distribution from the same family of distributions as the original variables, that family of distributions is said to be closed under convolution.
Examples of such univariate distributions are: Normal distribution, Poisson distribution, Binomial distribution (with common success probability), Negative binomial distribution (with common success probability), Gamma distribution(with common rate parameter), Chisquared distribution, Cauchy distribution, Hyperexponential distribution.
Examples:^{[2]}

 If X_{1} and X_{2} are Poisson random variables with means μ_{1} and μ_{2} respectively, then X_{1} + X_{2} is a Poisson random variable with mean μ_{1} + μ_{2}.
 The sum of gamma (n_{i}, β) random variables has a gamma (Σn_{i}, β) distribution.
 If X_{1} is a Cauchy (μ_{1}, σ_{1}) random variable and X_{2} is a Cauchy (μ_{2}, σ_{2}), then X_{1} + X_{2} is a Cauchy (μ_{1} + μ_{2}, σ_{1} + σ_{2}) random variable.
 If X_{1} and X_{2} are chisquared random variables with ν_{1} and ν_{2} degrees of freedom respectively, then X_{1} + X_{2} is a chisquared random variable with ν_{1} + ν_{2} degrees of freedom.
 If X_{1} is a normal (μ_{1}, σ_{1}^{2}) random variable and X_{2} is a normal (μ_{2}, σ_{2}^{2}) random variable, then X_{1} + X_{2} is a normal (μ_{1} + μ_{2}, σ_{1}^{2} + σ_{2}^{2}) random variable.
 The sum of N chisquared (1) random variables has a chisquared distribution with N degrees of freedom.
Other distributions are not closed under convolution, but their sum has a known distribution:
 The sum of n Bernoulli (p) random variables is a binomial (n, p) random variable.
 The sum of n geometric random variable with probability of success p is a negative binomial random variable with parameters n and p.
 The sum of n exponential (β) random variables is a gamma (n, β) random variable.
 The sum of the squares of N standard normal random variables has a chisquared distribution with N degrees of freedom.
Product of variables
The product of independent random variables X and Y may belong to the same family of distribution as X and Y: Bernoulli distribution and Lognormal distribution.
Example:
 If X_{1} and X_{2} are independent lognormal random variables with parameters (μ_{1}, σ_{1}^{2}) and (μ_{2}, σ_{2}^{2}) respectively, then X_{1} X_{2} is a lognormal random variable with parameters (μ_{1} + μ_{2}, σ_{1}^{2} + σ_{2}^{2}).
Minimum and maximum of independent random variables
For some distributions, the minimum value of several independent random variables is a member of the same family, with different parameters: Bernoulli distribution, Geometric distribution, Exponential distribution, Extreme value distribution, Pareto distribution, Rayleigh distribution, Weibull distribution.
Examples:
 If X_{1} and X_{2} are independent geometric random variables with probability of success p_{1} and p_{2} respectively, then min(X_{1}, X_{2}) is a geometric random variable with probability of success p = p_{1} + p_{2}  p_{1} p_{2}. The relationship is simpler if expressed in terms probability of failure: q = q_{1} q_{2}.
 If X_{1} and X_{2} are independent exponential random variables with mean μ_{1} and μ_{2} respectively, then min(X_{1}, X_{2}) is an exponential random variable with mean μ_{1} μ_{2}/(μ_{1} + μ_{2}).
Similarly, distributions for which the maximum value of several independent random variables is a member of the same family of distribution include: Bernoulli distribution, Power distribution.
Other
 If X and Y are independent standard normal random variables, X/Y is a Cauchy (0,1) random variable.
 If X_{1} and X_{2} are chisquared random variables with ν_{1} and ν_{2} degrees of freedom respectively, then (X_{1}/ν_{1})/(X_{2}/ν_{2}) is an F(ν_{1}, ν_{2}) random variable.
 If X is a standard normal random variable and U is a chisquared random variable with ν degrees of freedom, then \frac{X}{\sqrt{(U/\nu)}} is a Student's t (ν) random variable.
 If X_{1} is gamma (α_{1}, 1) random variable and X_{2} is a gamma (α_{2}, 1) random variable then X_{1}/(X_{1} + X_{2}) is a beta(α_{1}, α_{2}) random variable. More generally, if X_{1}is gamma(α_{1}, β_{1}) random variable and X_{2} is gamma(α_{2}, β_{2}) random variable then β_{2} X_{1}/(β_{2} X_{1} + β_{1} X_{2}) is a beta(α_{1}, α_{2}) random variable.
 If X and Y are exponential random variables with mean μ, then XY is a double exponential random variable with mean 0 and scale μ.
Approximate (limit) relationships
Approximate or limit relationship means
 either that the combination of an infinite number of iid random variables tends to some distribution,
 or that the limit when a parameter tends to some value approaches to a different distribution.
Combination of iid random variables:
 Given certain conditions, the sum (hence the average) of a sufficiently large number of iid random variables, each with finite mean and variance, will be approximately normally distributed.(This is central limit theorem (CLT)).
Special case of distribution parametrization:
 X is a Hypergeometric (m, N, n) random variable. If n and m are large compared to N, and p = m / N is not close to 0 or 1, then X approximately has a Binomial(n, p) Distribution.
 X is a betabinomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large, then X approximately has a binomial(n, p) distribution.
 If X is a binomial (n, p) random variable and if n is large and np is small then X approximately has a Poisson(np) distribution.
 If X is a negative binomial random variable with r large, P near 1, and r(1P) = λ, then X approximately has a Poisson distribution with mean λ.
Consequences of the CLT:
 If X is a Poisson random variable with large mean, then for integers j and k, P(j ≤ X ≤ k) approximately equals to P(j  1/2 ≤ Y ≤ k + 1/2) where Y is a normal distribution with the same mean and variance as X.
 If X is a binomial(n, p) random variable with large n and np, then for integers j and k, P(j ≤ X ≤ k) approximately equals to P(j  1/2 ≤ Y ≤ k + 1/2) where Y is a normal random variable with the same mean and variance as X, i. e. np and np(1p).
 If X is a beta random variable with parameters α and β equal and large, then X approximately has a normal distribution with the same mean and variance, i. e. mean α/(α + β) and variance αβ/((α+β)^{2}(α + β + 1)).
 If X is a gamma(α, β) random variable and the shape parameter α is large relative to the scale parameter β, then X approximately has a normal random variable with the same mean and variance.
 If X is a Student's t random variable with a large number of degrees of freedom ν then X approximately has a standard normal distribution.
 If X is an F(ν, ω) random variable with ω large, then ν X is approximately distributed As a chisquared random variable with ν degrees of freedom.
Compound (or Bayesian) relationships
When one or more parameter(s) of a distribution are random variables, the compound distribution is the marginal distribution of the variable.
Examples:
 If XN is a binomial (N,p) random variable, where parameter N is a random variable with negativebinomial (m, r) distribution, then X is distributed as a negativebinomial (m, r/(p+qr)).
 If XN is a binomial (N,p) random variable, where parameter N is a random variable with Poisson (μ) distribution, then X is distributed as a Poisson (μp).
 If Xμ is a Poisson (μ) random variable and parameter μ is random variable with gamma (m, β) distribution, then X is distributed as a negativebinomial (m, μβ/(μ+β)), sometimes called GammaPoisson distribution if m is not integer.
Some distributions have been specially named as compounds: BetaBinomial distribution, BetaPascal distribution, GammaNormal distribution.
Examples:
 If X is a Binomial (n,p) random variable, and parameter p is a random variable with beta (α, β) distribution, then X is distributed as a BetaBinomial(α, β,n).
 If X is a negativebinomial (m,p) random variable, and parameter p is a random variable with beta (α, β) distribution, then X is distributed as a BetaPascal(α, β,m).
See also
References
External links
 Interactive graphic: Univariate Distribution Relationships
