Relationships among probability distributions

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 (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:

• 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).

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.

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 has cumulative distribution function F_X, 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 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.

• 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 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.

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.

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 (α_i, β) random variables has a gamma (Σα_i, 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. 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:

• 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).

(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:

• 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 + p₂ − p₁ 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.

• 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 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 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 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.)