Relationships among probability distributions

These relations can be categorized in the following groups: Multiplying the variable by any positive real constant yields a scaling of the original distribution.

Example: The affine transform ax + b yields a relocation and scaling of the original distribution.

Examples: Some distributions are invariant under a specific transformation.

Example: The distribution of the sum of independent random variables is the convolution of their distributions.

independent random variables

each with probability mass functions

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.

Approximate or limit relationship means Combination of iid random variables: Special case of distribution parametrization: Consequences of the CLT: When one or more parameter(s) of a distribution are random variables, the compound distribution is the marginal distribution of the variable.

Relationships among some of univariate probability distributions are illustrated with connected lines. dashed lines means approximate relationship. more info: [ 1 ]
Relationships between univariate probability distributions in ProbOnto . [ 2 ]