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

Special case of distribution parametrization

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: **

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: **

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: **

Other cases

Some distributions are invariant under a specific transformation. **Example: **

Functions of several variables

Sum of variables

The distribution of the sum of independent random variables is the convolution of their distributions. Suppose Z is the sum of n independent random variables each with probability mass functions f_{X_i}(x). Then 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: Other distributions are not closed under convolution, but their sum has a known distribution:

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: **

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: ** 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

Approximate (limit) relationships

Approximate or limit relationship means Combination of iid random variables: **Special case of distribution parametrization: ** Consequences of the CLT:

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: ** Some distributions have been specially named as compounds: beta-binomial distribution, Beta negative binomial distribution, gamma-normal distribution. **Examples: **