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Relationships among probability distributions

Topic in probability theory and statistics

Inprobability 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.

• Abinomial distribution with parameters n = 1 and p is a Bernoulli distribution with parameter p.
• Anegative binomial distribution with parameters n = 1 and p is a geometric distribution with parameter p.
• Agamma distribution with shape parameter α = 1 and rate parameter β is an exponential distribution with rate parameter β.
• Agamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom.
• Achi-squared distribution with 2 degrees of freedom (k = 2) is an exponential distribution with a mean value of 2 (rate λ = 1/2 .)
• AWeibull distribution with shape parameter k = 1 and rate parameter β is an exponential distribution with rate parameter β.
• Abeta distribution with shape parameters α = β = 1 is a continuous uniform distribution over the real numbers 0 to 1.
• Abeta-binomial distribution with parameter n and shape parameters α = β = 1 is a discrete uniform distribution over the integers 0 to n.
• AStudent's t-distribution with one degree of freedom (v = 1) is a Cauchy distribution with location parameter x = 0 and scale parameter γ = 1.
• ABurr distribution with parameters c = 1 and k (and scale λ) is a Lomax distribution with shape k (and scale λ.)

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:

• IfX 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).

• IfX is a gamma random variable with shape and scale parameters (k, θ), then Y = aX is a gamma random variable with parameters (k,aθ).

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:

• IfZ 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).

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.
• IfX is an F(ν₁, ν₂) random variable then 1/X is an F(ν₂, ν₁) random variable.

Some distributions are invariant under a specific transformation.

Example:

• IfX is a beta (α, β) random variable then (1 − X) is a beta (β, α) random variable.
• IfX is a binomial (n, p) random variable then (n − X) is a binomial (n, 1 − p) random variable.
• IfX has cumulative distribution function F_X, then the inverse of the cumulative distribution F
  _X(X) is a standard uniform (0,1) random variable
• IfX is a normal (μ, σ²) random variable then e^X 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.

• IfX 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 chi-squared distribution with one degree of freedom.
• IfX is a Student’s t random variable with ν degree of freedom, then X² is an F (1,ν) random variable.
• IfX is a double exponential random variable with mean 0 and scale λ, then |X| is an exponential random variable with mean λ.
• Ageometric random variable is the floor of an exponential random variable.
• Arectangular random variable is the floor of a uniform random variable.
• Areciprocal random variable is the exponential of a uniform random variable.

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

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

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. Often (always?) these distributions are also stable distributions (see also Discrete-stable distribution).

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]

• • IfX₁ 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 (α_i, β) random variables has a gamma (Σα_i, β) distribution.
  • IfX₁ 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.
  • IfX₁ 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. Since n is an integer, the gamma distribution is also a Erlang distribution.
• The sum of the squares of N standard normal random variables has a chi-squared distribution with N degrees of freedom.

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:

• IfX₁ 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).

(See also Product distribution.)

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:

• IfX₁ 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 + p₂ − p₁ p₂. The relationship is simpler if expressed in terms probability of failure: q = q₁ q₂.
• IfX₁ 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.

• IfX and Y are independent standard normal random variables, X/Y is a Cauchy (0,1) random variable.
• IfX₁ 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.
• IfX 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.
• IfX₁ is a gamma (α₁, 1) random variable and X₂ is an independent gamma (α₂, 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 β₂X₁/(β₂ X₁ + β₁ X₂) is a beta(α₁, α₂) random variable.
• IfX 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 X_i are independent Bernoulli random variables then their parity (XOR) is a Bernoulli variable described by the piling-up lemma.

(See also ratio distribution.)

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.
• IfX is a binomial (n, p) random variable and if n is large and np is small then X approximately has a Poisson(np) distribution.
• IfX 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:

• IfX 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.
• IfX is a binomial(n, p) random variable with large np and n(1 − p), 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(1 − p).
• IfX 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)).
• IfX 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.
• IfX is a Student's t random variable with a large number of degrees of freedom ν then X approximately has a standard normal distribution.
• IfX 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.

When one or more parameter(s) of a distribution are random variables, the compound distribution is the marginal distribution of the variable.

Examples:

• IfX | 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)).
• IfX | 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).
• IfX | 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 negative binomial distribution, gamma-normal distribution.

Examples:

• IfX 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).
• IfX is a negative-binomial(r,p) random variable, and parameter p is a random variable with beta(α,β) distribution, then X is distributed as a Beta negative binomial distribution(r,α,β).

• Central limit theorem
• Compound probability distribution
• List of convolutions of probability distributions

• Interactive graphic: Univariate Distribution Relationships
• ProbOnto - Ontology and knowledge base of probability distributions: ProbOnto
• Probability Distributome project includes calculators, simulators, experiments, and navigators for inter-distributional refashions and distribution meta-data.