# Relationships among probability distributions

Relationships among some of univariate probability distributions are illustrated with connected lines. dashed lines means approximate relationship. more info:[1] Note: The transformation from N(mu, sigma^2) to N(0, 1) should be (X - mu)/sigma, not (X - mu)/sigma^2
Relationships between univariate probability distributions in ProbOnto.[2]

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[clarification needed];
• Conjugate priors.

## 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 (r, λ), then Y = aX is a gamma random variable with parameters (r,λ/a).
• If X is a gamma random variable with shape and scale parameters (α, β), then Y = aX is a gamma random variable with parameters (α,).

### 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 (nX) is a binomial (n, 1 − p) random variable.
• If X has cumulative distribution function FX, then the inverse of the cumulative distribution F
X
(X) is a standard uniform (0,1) random variable
• 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

The distribution of the sum of independent random variables is the convolution of their distributions. Suppose ${\displaystyle Z}$ is the sum of ${\displaystyle n}$ independent random variables ${\displaystyle X_{1},\dots ,X_{n}}$ each with probability mass functions ${\displaystyle f_{X_{i}}(x)}$. Then

${\displaystyle Z=\sum _{i=1}^{n}{X_{i}}}$

has

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 distributions, Poisson distributions, binomial distributions (with common success probability), negative binomial distributions (with common success probability), gamma distributions (with common rate parameter), chi-squared distributions, Cauchy distributions, hyperexponential distributions.

Examples:[3][4]

• 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.
• The sum of gamma (ni, β) random variables has a gamma ni, β) distribution.
• If X1 is a Cauchy (μ1, σ1) random variable and X2 is a Cauchy (μ2, σ2), then X1 + X2 is a Cauchy (μ1 + μ2, σ1 + σ2) random variable.
• If X1 and X2 are chi-squared random variables with ν1 and ν2 degrees of freedom respectively, then X1 + X2 is a chi-squared random variable with ν1 + ν2 degrees of freedom.
• If X1 is a normal (μ1, σ2
1
) random variable and X2 is a normal (μ2, σ2
2
) random variable, then X1 + X2 is a normal (μ1 + μ2, σ2
1
+ σ2
2
) random variable.
• The sum of N chi-squared (1) random variables has a chi-squared 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 variables 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.
• If the exponential random variables have a common rate parameter, their sum has an Erlang distribution, a special case of the gamma distribution.
• The sum of the squares of N standard normal random variables has a chi-squared 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 log-normal distribution.

Example:

• If X1 and X2 are independent log-normal random variables with parameters (μ1, σ2
1
) and (μ2, σ2
2
) respectively, then X1 X2 is a log-normal random variable with parameters (μ1 + μ2, σ2
1
+ σ2
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 X1 and X2 are independent 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 + p2p1 p2. The relationship is simpler if expressed in terms probability of failure: q = q1 q2.
• If X1 and X2 are independent exponential random variables with rate μ1 and μ2 respectively, then min(X1, X2) is an exponential random variable with rate μ = μ1 + μ2.

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 law distribution.

### Other

• If X and Y are independent standard normal random variables, X/Y is a Cauchy (0,1) random variable.
• If X1 and X2 are independent chi-squared random variables with ν1 and ν2 degrees of freedom respectively, then (X1/ν1)/(X2/ν2) is an F(ν1, ν2) random variable.
• If X is a standard normal random variable and U is an independent chi-squared random variable with ν degrees of freedom, then ${\displaystyle {\frac {X}{\sqrt {(U/\nu )}}}}$ is a Student's t(ν) random variable.
• If X1 is a gamma (α1, 1) random variable and X2 is an independent gamma2, 1) random variable then X1/(X1 + X2) is a beta(α1, α2) random variable. More generally, if X1 is a gamma(α1, β1) random variable and X2 is an independent gamma(α2, β2) random variable then β2 X1/(β2 X1 + β1 X2) is a beta(α1, α2) random variable.
• If X and Y are independent exponential random variables with mean μ, then X − Y is a double exponential random variable with mean 0 and scale μ.
• If Xi are independent Bernoulli random variables then their parity (XOR) is a Bernoulli variable described by the piling-up lemma.

## 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 the 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 beta-binomial 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(1 − P) = 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(jXk) approximately equals to P(j − 1/2 ≤ Yk + 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 np and n(1 − p), then for integers j and k, P(jXk) approximately equals to P(j − 1/2 ≤ Yk + 1/2) where Y is a normal random variable with the same mean and variance as X, i.e. np and np(1 − p).
• 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 νX is approximately distributed as a chi-squared 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 X | N is a binomial (N,p) random variable, where parameter N is a random variable with negative-binomial (m, r) distribution, then X is distributed as a negative-binomial (m, r/(p + qr)).
• If X | N 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 X | μ is a Poisson(μ) random variable and parameter μ is random variable with gamma(m, θ) distribution (where θ is the scale parameter), then X is distributed as a negative-binomial (m, θ/(1 + θ)), sometimes called gamma-Poisson distribution.

Some distributions have been specially named as compounds: beta-binomial distribution, beta-Pascal distribution, gamma-normal 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 Beta-Binomial(α,β,n).
• If X is a negative-binomial(m,p) random variable, and parameter p is a random variable with beta(α,β) distribution, then X is distributed as a Beta-Pascal(α,β,m).