Relationships
In all statements about two random variables, the random variables are implicitly independent.
Geometric / negative binomial: If each Xi is geometric random variable with probability of success p then the sum of n Xi's is a negative binomial random variable with parameters n and p.
Negative binomial / geometric: A negative binomial distribution with r = 1 is a geometric distribution.
Negative binomial / Poisson: If X has a negative binomial random variable with r large, p near 1, and r(1-p) = λ, then FX ≈ FY where Y is a Poisson random variable with mean λ.
Beta-binomial / discrete uniform: A beta-binomial (n, 1, 1) random variable is a discrete uniform random variable over the values 0 ... n.
Beta-binomial / binomial: Let X be a beta-binomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large. If Y is a binomial(n, p) random variable then FX ≈ FY.
Hypergeometric / binomial: The difference between a hypergeometric distribution and a binomial distribution is the difference between sampling without replacement and sampling with replacement. As the population size increases relative to the sample size, the difference becomes negligible.
Geometric / geometric: If X1 and X2 are 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 in terms of failure probabilities: q = q1 q2.
Poisson / Poisson: If X1 and X2 are Poisson random variables with means μ1 and μ2respectively, then X1 + X2 is a Poisson random variable with mean μ1 + μ2.
Binomial / Poisson: If X is a binomial(n, p) random variable and Y is a Poisson(np) distribution then P(X = n) ≈ P(Y = n) if n is large and np is small. For more information, see
Poisson approximation to binomial.
Binomial / Bernoulli: If X is a binomial(n, p) random variable with n = 1, X is a Bernoulli(p) random variable.
Bernoulli / Binomial: The sum of n Bernoulli(p) random variables is a binomial(n, p) random variable.
Poisson / normal: If X is a Poisson random variable with large mean and Y is a normal distribution with the same mean and variance as X, then for integers j and k, P(j ≤ X ≤ k) ≈ P(j - 1/2 ≤ Y ≤ k + 1/2). For more information, see
normal approximation to Poisson.
Binomial / normal: If X is a binomial(n, p) random variable and Y is a normal random variable with the same mean and variance as X, i.e. np and np(1-p), then for integers j and k, P(j ≤ X ≤ k) ≈ P(j - 1/2 ≤ Y ≤ k + 1/2). The approximation is better when p ≈ 0.5 and when n is large. For more information, see
normal approximation to binomial.
Lognormal / lognormal: If X1 and X2 are lognormal random variables with parameters (μ1, σ12) and (μ2, σ22) respectively, then X1 X2 is a lognormal random variable with parameters (μ1 + μ2, σ12 + σ22).
Normal / lognormal: If X is a normal (μ, σ2) random variable then eX 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.
Beta / normal: If X is a beta random variable with parameters α and β equal and large, FX ≈ FY where Y is a normal random variable with the same mean and variance as X, i.e. mean α/(α + β) and variance αβ/((α+β)2(α + β + 1)). For more information, see
normal approximation to beta.
Normal / standard normal: If X is a normal(μ, σ2) random variable then (X - μ)/σ is a standard normal random variable. Conversely, If X is a normal(0,1) random variable then σ X + μ is a normal (μ, σ2) random variable.
Normal / normal: If X1 is a normal (μ1, σ12) random variable and X2 is a normal (μ2, σ22) random variable, then X1 + X2 is a normal (μ1 + μ2, σ12 + σ22) random variable.
Gamma / normal: If X is a gamma(α, β) random variable and Y is a normal random variable with the same mean and variance as X, then FX ≈ FY if the shape parameter α is large relative to the scale parameter β. For more information, see
normal approximation to gamma.
Gamma / beta: If X1 is gamma(α1, 1) random variable and X2 is a gamma (α2, 1) random variable then X1/(X1 + X2) is a beta(α1, α2) random variable. More generally, if X1 is gamma(α1, β1) random variable and X2 is gamma(α2, β2) random variable then β2 X1/(β2 X1 + β1 X2) is a beta(α1, α2) random variable.
Beta / uniform: A beta random variable with parameters α = β = 1 is a uniform random variable.
Chi-squared / chi-squared: 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.
Standard normal / chi-squared: The square of a standard normal random variable has a chi-squared distribution with one degree of freedom. The sum of the squares of n standard normal random variables is has a chi-squared distribution with n degrees of freedom.
Gamma / chi-squared: If X is a gamma (α, β) random variable with α = ν/2 and β = 2, then X is a chi-squared random variable with ν degrees of freedom.
Cauchy / standard normal: If X and Y are standard normal random variables, X/Y is a Cauchy(0,1) random variable.
Student t / standard normal: If X is a t random variable with a large number of degrees of freedom ν then FX ≈ FY where Y is a standard normal random variable. For more information, see
normal approximation to t.
Snedecor F / chi-squared: If X is an F(ν, ω) random variable with ω large, then ν X is approximately distributed as a chi-squared random variable with ν degrees of freedom.
Chi-squared / Snedecor F: If X1 and X2 are chi-squared random variables with ν1 and ν2 degrees of freedom respectively, then (X1/ν1)/(X2/ν2) is an F(ν1, ν2) random variable.
Chi-squared / exponential: A chi-squared distribution with 2 degrees of freedom is an exponential distribution with mean 2.
Exponential / chi-squared: An exponential random variable with mean 2 is a chi-squared random variable with two degrees of freedom.
Gamma / exponential: The sum of n exponential(β) random variables is a gamma(n, β) random variable.
Exponential / gamma: A gamma distribution with shape parameter α = 1 and scale parameter β is an exponential(β) distribution.
Exponential / uniform: If X is an exponential random variable with mean λ, then exp(-X/λ) is a uniform random variable. More generally, sticking any random variable into its CDF yields a uniform random variable.
Uniform / exponential: If X is a uniform random variable, -λ log X is an exponential random variable with mean λ. More generally, applying the inverse CDF of any random variable X to a uniform random variable creates a variable with the same distribution as X.
Cauchy reciprocal: 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.
Cauchy sum: 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.
Student t / Cauchy: A random variable with a t distribution with one degree of freedom is a Cauchy(0,1) random variable.
Student t / Snedecor F: If X is a t random variable with ν degree of freedom, then X2 is an F(1,ν) random variable.
Snedecor F / Snedecor F: If X is an F(ν1, ν2) random variable then 1/X is an F(ν2, ν1) random variable.
Exponential / Exponential: If X1 and X2 are exponential random variables with mean μ1 and μ2 respectively, then min(X1, X2) is an exponential random variable with mean μ1 μ2/(μ1 + μ2).
Exponential / Weibull: If X is an exponential random variable with mean β, then X1/γis a Weibull(γ, β) random variable.
Weibull / Exponential: If X is a Weibull(1, β) random variable, X is an exponential random variable with mean β.
Exponential / Double exponential: If X and Y are exponential random variables with mean μ, then X-Y is a double exponential random variable with mean 0 and scale μ
Double exponential / exponential: If X is a double exponential random variable with mean 0 and scale λ, then |X| is an exponential random variable with mean λ.