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Concept
Relationships among probability distributions
Summary
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;
Conjugate priors.
A binomial distribution with parameters n = 1 and p is a Bernoulli distribution with parameter p.
A negative binomial distribution with parameters n = 1 and p is a geometric distribution with parameter p.
A gamma distribution with shape parameter α = 1 and rate parameter β is an exponential distribution with rate parameter β.
A gamma distribution with shape parameter α = v/2 and rate parameter β = 1/2 is a chi-squared distribution with ν degrees of freedom.
A chi-squared distribution with 2 degrees of freedom (k = 2) is an exponential distribution with a mean value of 2 (rate λ = 1/2 .)
A Weibull distribution with shape parameter k = 1 and rate parameter β is an exponential distribution with rate parameter β.
A beta distribution with shape parameters α = β = 1 is a continuous uniform distribution over the real numbers 0 to 1.
A beta-binomial distribution with parameter n and shape parameters α = β = 1 is a discrete uniform distribution over the integers 0 to n.
A Student's t-distribution with one degree of freedom (v = 1) is a Cauchy distribution with location parameter x = 0 and scale parameter γ = 1.
A Burr 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.
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