Relationships among probability distributions

• One distribution is a special case of another with a broader parameter space
• Transforms ;
• Combinations ;
• Approximation relationships;
• Compound relationships ;
• Duality;
• Conjugate priors.

  Special case of distribution parametrization

• A binomial random variable with n = 1, is a Bernoulli random variable.
• A negative binomial distribution with n = 1 is a geometric distribution.
• A gamma distribution with shape parameter α = 1 and scale parameter θ is an exponential distribution with expected value θ.
• A gamma random variable with α = ν/2 and β = 1/2, is a chi-squared random variable with ν degrees of freedom.
• A chi-squared distribution with 2 degrees of freedom is an exponential distribution with mean 2 and vice versa.
• A Weibull random variable is an exponential random variable with mean β.
• A beta random variable with parameters α = β = 1 is a uniform random variable.
• A beta-binomial random variable is a discrete uniform random variable over the values 0,..., n.
• A random variable with a t distribution with one degree of freedom is a Cauchy random variable.
• When c = 1, the Burr type XII distribution becomes the Pareto Type II distribution.

  Transform of a variable

Multiple of a random variable

• If X is a gamma random variable with shape and rate parameters, then Y = aX is a gamma random variable with parameters.
• If X is a gamma random variable with shape and scale parameters, then Y = aX is a gamma random variable with parameters.

  Linear function of a random variable

• If Z is a normal random variable with parameters, then X = aZ + b is a normal random variable with parameters.

  Reciprocal of a random variable

• If X is a Cauchy random variable, then 1/X is a Cauchy random variable where C = μ² + σ².
• If X is an F random variable then 1/X is an F random variable.

  Other cases

• If X is a beta random variable then is a beta random variable.
• If X is a binomial random variable then is a binomial random variable.
• If X has cumulative distribution function F_X, then the inverse of the cumulative distribution F is a standard uniform random variable
• If X is a normal random variable then e^X is a lognormal random variable.
• If X is an exponential random variable with mean β, then X^1/γ is a Weibull 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 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.

  Functions of several variables

Sum of variables

• *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 random variables has a gamma distribution.
• *If X₁ is a Cauchy random variable and X₂ is a Cauchy, then X₁ + X₂ is a Cauchy 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 random variable and X₂ is a normal random variable, then X₁ + X₂ is a normal random variable.
• *The sum of N chi-squared random variables has a chi-squared distribution with N degrees of freedom.

• The sum of n Bernoulli random variables is a binomial random variable.
• The sum of n geometric random variable with probability of success p is a negative binomial random variable with parameters n and p.
• The sum of n exponential random variables is a gamma random variable.
• *If the exponential random variables have a common rate parameter, their sum has an Erlang distribution, a special case of the gamma distribution.
• The sum of the squares of N standard normal random variables has a chi-squared''' distribution with N degrees of freedom.

  Product of variables

• If X₁ and X₂ are independent log-normal random variables with parameters and respectively, then X₁ X₂ is a log-normal random variable with parameters.

  Minimum and maximum of independent random variables

• If X₁ and X₂ are independent geometric random variables with probability of success p₁ and p₂ respectively, then min 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 is an exponential random variable with rate μ = μ₁ + μ₂.

Other

• If X and Y are independent standard normal random variables, X/Y is a Cauchy random variable.
• If X₁ and X₂ are independent chi-squared random variables with ν₁ and ν₂ degrees of freedom respectively, then / is an F 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 is a Student's t random variable.
• If X₁ is a gamma random variable and X₂ is an independent gamma random variable then X₁/ is a beta random variable. More generally, if X₁ is a gamma random variable and X₂ is an independent gamma random variable then β₂ X₁/ is a beta 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.

  Approximate (limit) relationships

• 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.

• Given certain conditions, the sum 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.

• X is a hypergeometric 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 distribution.
• X is a beta-binomial random variable with parameters. Let p = α/ and suppose α + β is large, then X approximately has a binomial distribution.
• If X is a binomial random variable and if n is large and np is small then X approximately has a Poisson distribution.
• If X is a negative binomial random variable with r large, P near 1, and r = λ, then X approximately has a Poisson distribution with mean λ.

• If X is a Poisson random variable with large mean, then for integers j and k, P approximately equals to P where Y is a normal distribution with the same mean and variance as X.
• If X is a binomial random variable with large np and n, then for integers j and k, P approximately equals to P where Y is a normal random variable with the same mean and variance as X, i.e. np and np.
• If X 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/ and variance αβ/²).
• If X is a gamma 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.
• If X is a Student's t random variable with a large number of degrees of freedom ν then X approximately has a standard normal distribution.
• If X is an F random variable with ω large, then νX is approximately distributed as a chi-squared random variable with ν degrees of freedom.

  Compound (or Bayesian) relationships

• If X | N is a binomial random variable, where parameter N is a random variable with negative-binomial distribution, then X is distributed as a negative-binomial.
• If X | N is a binomial random variable, where parameter N is a random variable with Poisson distribution, then X is distributed as a Poisson.
• If X | μ is a Poisson random variable and parameter μ is random variable with gamma distribution, then X is distributed as a negative-binomial, sometimes called gamma-Poisson distribution.

• If X is a Binomial random variable, and parameter p is a random variable with beta distribution, then X is distributed as a Beta-Binomial.
• If X is a negative-binomial random variable, and parameter p is a random variable with beta distribution, then X is distributed as a Beta-Pascal.