Probability distributions have a surprising number inter-connections. A dashed line in the chart below indicates an approximate (limit) relationship between two distribution families. A solid line indicates an exact relationship: special case, sum, or transformation.

**Click on a distribution** for the parameterization of that distribution. **Click on an arrow** for details on the relationship represented by the arrow.

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The chart above is adapted from the chart originally published by Lawrence Leemis in 1986 (Relationships Among Common Univariate Distributions, American Statistician 40:143-146.) Leemis published a larger chart in 2008 which is available online.

If you would like for me to do a one-day seminar explaining in detail the information in this chart, please let me know.

## Parameterizations

The precise relationships between distributions depend on parameterization. The relationships detailed below depend on the following parameterizations for the PDFs.

Let *C*(*n*, *k*) denote the binomial coefficient(*n*, *k*) and *B*(*a*, *b*) = Γ(*a*) Γ(*b*) / Γ(*a* + *b*).

**Geometric**: *f*(*x*) = *p* (1-*p*)* ^{x}* for non-negative integers

*x*.

**Discrete uniform**: *f*(*x*) = 1/*n* for *x* = 1, 2, …, *n*.

**Negative binomial**: *f*(*x*) = *C*(*r* + *x* – 1, *x*) *p ^{r}*(1-

*p*)

*for non-negative integers*

^{x}*x*. See notes on the negative binomial distribution.

**Beta binomial**: *f*(*x*) = *C*(*n*, *x*) *B*(α + *x*, *n* + β – *x*) / *B*(α, β) for *x* = 0, 1, …, *n*.

**Hypergeometric**: *f*(*x*) = *C*(*M*, *x*) *C*(*N*–*M*, *K* – *x*) / *C*(N, *K*) for *x* = 0, 1, …, *N*.

**Poisson**: *f(x*) = exp(-λ) λ* ^{x}*/

*x*! for non-negative integers

*x*. The parameter λ is both the mean and the variance.

**Binomial**: *f*(*x*) = *C*(*n*, *x*) *p ^{x}*(1 –

*p*)

^{n–x}for

*x*= 0, 1, …,

*n*.

**Bernoulli**: *f*(*x*) = *p ^{x}*(1 –

*p*)

^{1-x}where

*x*= 0 or 1.

**Lognormal**: *f*(*x*) = (2πσ^{2})^{-1/2} exp( -(log(*x*) – μ)^{2}/ 2σ^{2}) / *x* for positive *x*. Note that μ and σ^{2} are not the mean and variance of the distribution.

**Normal **: *f*(*x*) = (2π σ^{2})^{-1/2} exp( – ½((*x* – μ)/σ)^{2} ) for all *x*.

**Beta**: *f*(*x*) = Γ(α + β) *x*^{α-1}(1 –* x*)^{β-1} / (Γ(α) Γ(β)) for 0 ≤ *x* ≤ 1.

**Standard normal**: *f*(*x*) = (2π)^{-1/2} exp( –*x*^{2}/2) for all *x*.

**Chi-squared**: *f*(*x*) = *x*^{-ν/2-1} exp(-*x*/2) / Γ(ν/2) 2^{ν/2} for positive x. The parameter ν is called the degrees of freedom.

**Gamma**: *f*(*x*) = β^{-α} *x*^{α-1} exp(-*x*/β) / Γ(α) for positive x. The parameter α is called the shape and β is the scale.

**Uniform**: *f*(*x*) = 1 for 0 ≤ *x* ≤ 1.

**Cauchy**: *f*(*x*) = σ/(π( (*x* – μ)^{2} + σ^{2}) ) for all *x*. Note that μ and σ are location and scale parameters. Mean and variance are undefined for the Cauchy distribution.

**Snedecor F**:

*f*(

*x*) is proportional to

*x*

^{(ν1 – 2)/2}/ (1 + (ν

_{1}/ν

_{2})

*x*)

^{(ν1 + ν2)/2}for positive

*x*.

**Exponential**: *f*(*x*) = exp(-*x*/μ)/μ for positive *x*. The parameter μ is the mean.

**Student t**:

*f*(

*x*) is proportional to (1 + (

*x*

^{2}/ν))

^{-(ν + 1)/2 }for positive

*x*. The parameter ν is called the degrees of freedom.

**Weibull**: *f*(*x*) = (γ/β) *x*^{γ-1} exp(- *x*^{γ}/β) for positive *x*. The parameter γ is the shape and β is the scale.

**Double exponential **: *f*(*x*) = exp(-|*x*-μ|/σ) / 2σ for all *x*. The parameter μ is the location and mean; σ is the scale. For comparison, see distribution parameterizations in R/S-PLUS and Mathematica.

## Relationships

In all statements about two random variables, the random variables are implicitly independent.

**Geometric / negative binomial**: If each *X _{i}* is geometric random variable with probability of success

*p*then the sum of

*n*

*X*‘s is a negative binomial random variable with parameters

_{i}*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 *F _{X}* ≈

*F*where

_{Y}*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 *F _{X}* ≈

*F*.

_{Y}**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 *X*_{1} and *X*_{2} are geometric random variables with probability of success *p*_{1} and *p*_{2} respectively, then min(*X*_{1}, *X*_{2}) is a geometric random variable with probability of success *p* = *p*_{1} + *p*_{2} – *p*_{1} *p*_{2}. The relationship is simpler in terms of failure probabilities: *q* = *q*_{1} *q*_{2}.

**Poisson / Poisson**: If *X*_{1} and *X*_{2} are Poisson random variables with means μ_{1} and μ_{2} respectively, then *X*_{1} + *X*_{2} is a Poisson random variable with mean μ_{1} + μ_{2}.

**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 *X*_{1} and *X*_{2} are lognormal random variables with parameters (μ_{1}, σ_{1}^{2}) and (μ_{2}, σ_{2}^{2}) respectively, then *X*_{1} *X*_{2} is a lognormal random variable with parameters (μ_{1} + μ_{2}, σ_{1}^{2} + σ_{2}^{2}).

**Normal / lognormal**: If *X* is a normal (μ, σ^{2}) random variable then *e ^{X}* is a lognormal (μ, σ

^{2}) random variable. Conversely, if

*X*is a lognormal (μ, σ

^{2}) random variable then log

*X*is a normal (μ, σ

^{2}) random variable.

**Beta / normal**: If *X* is a beta random variable with parameters α and β equal and large, *F _{X}* ≈

*F*where

_{Y}*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, σ^{2}) 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, σ^{2}) random variable.

**Normal / normal**: If *X*_{1} is a normal (μ_{1}, σ_{1}^{2}) random variable and *X*_{2} is a normal (μ_{2}, σ_{2}^{2}) random variable, then *X*_{1} + *X*_{2} is a normal (μ_{1} + μ_{2}, σ_{1}^{2} + σ_{2}^{2}) 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 *F _{X}* ≈

*F*if the shape parameter α is large relative to the scale parameter β. For more information, see normal approximation to gamma.

_{Y}**Gamma / beta**: If *X*_{1} is gamma(α_{1}, 1) random variable and *X*_{2} is a gamma (α_{2}, 1) random variable then *X*_{1}/(*X*_{1} + *X*_{2}) is a beta(α_{1}, α_{2}) random variable. More generally, if *X*_{1} is gamma(α_{1}, β_{1}) random variable and *X*_{2} is gamma(α_{2}, β_{2}) random variable then β_{2} *X*_{1}/(β_{2} *X*_{1} + β_{1} *X*_{2}) is a beta(α_{1}, α_{2}) random variable.

**Beta / uniform**: A beta random variable with parameters α = β = 1 is a uniform random variable.

**Chi-squared / chi-squared**: If *X*_{1} and* X*_{2} are chi-squared random variables with ν_{1} and ν_{2} degrees of freedom respectively, then *X*_{1} + *X*_{2} is a chi-squared random variable with ν_{1} + ν_{2} 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

*F*≈

_{X}*F*where

_{Y}*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

*X*

_{1}and

*X*

_{2}are chi-squared random variables with ν

_{1}and ν

_{2}degrees of freedom respectively, then (

*X*

_{1}/ν

_{1})/(

*X*

_{2}/ν

_{2}) is an

*F*(ν

_{1}, ν

_{2}) 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* = μ^{2} + σ^{2}.

**Cauchy sum**: If *X*_{1} is a Cauchy (μ_{1}, σ_{1}) random variable and *X*_{2} is a Cauchy (μ_{2}, σ_{2}), then *X*_{1} + *X*_{2} is a Cauchy (μ_{1} + μ_{2}, σ_{1} + σ_{2}) 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

*X*

^{2}is an

*F*(1,ν) random variable.

**Snedecor F / Snedecor F**: If

*X*is an

*F*(ν

_{1}, ν

_{2}) random variable then 1/

*X*is an

*F*(ν

_{2}, ν

_{1}) random variable.

**Exponential / Exponential**: If *X*_{1} and *X*_{2} are exponential random variables with mean μ_{1} and μ_{2} respectively, then min(*X*_{1}, *X*_{2}) is an exponential random variable with mean μ_{1} μ_{2}/(μ_{1} + μ_{2}).

**Exponential / Weibull**: If *X* is an exponential random variable with mean β, then *X*^{1/γ} is 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.