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.

## Special case of distribution parametrization

- A binomial (
*n*,*p*) random variable with*n*= 1, is a Bernoulli (*p*) 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 (
*α*,*β*) 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 (1,
*β*) 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 (
*n*, 1, 1) 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(0,1) random variable.

## Transform of a variable

### Multiple of a random variable

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.

**Example: **

- If
*X*is a gamma random variable with shape and rate parameters (*r*,*λ*), then*Y*=*aX*is a gamma random variable with parameters (*r*,*λ*/*a*).

- If
*X*is a gamma random variable with shape and scale parameters (*α*,*β*), then*Y*=*aX*is a gamma random variable with parameters (*α*,*aβ*).

### Linear function of a random variable

The affine transform *ax* + *b* yields a **relocation and scaling** of the original distribution. The following are self-replicating:
Normal distribution, Cauchy distribution, Logistic distribution, Error distribution, Power distribution, Rayleigh distribution.

**Example: **

- If
*Z*is a normal random variable with parameters (*μ*=*m*,*σ*^{2}=*s*^{2}), then*X*=*aZ*+*b*is a normal random variable with parameters (*μ*=*am*+*b*,*σ*^{2}=*a*^{2}*s*^{2}).

### Reciprocal of a random variable

The reciprocal 1/*X* of a random variable *X*, is a member of the same family of distribution as *X*, in the following cases:
Cauchy distribution, F distribution, log logistic distribution.

**Examples: **

- 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}. - If
*X*is an*F*(*ν*_{1},*ν*_{2}) random variable then 1/*X*is an*F*(*ν*_{2},*ν*_{1}) random variable.

### Other cases

Some distributions are invariant under a specific transformation.

**Example: **

- If
*X*is a**beta**(*α*,*β*) random variable then (1 −*X*) is a**beta**(*β*,*α*) random variable. - If
*X*is a**binomial**(*n*,*p*) random variable then (*n*−*X*) is a**binomial**(*n*, 1 −*p*) random variable. - If
*X*has cumulative distribution function*F*_{X}, then the inverse of the cumulative distribution*F*^{}_{X}(*X*) is a standard**uniform**(0,1) random variable - 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.

- If
*X*is an**exponential**random variable with mean*β*, then*X*^{1/γ}is a**Weibull**(*γ*,*β*) 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*X*^{2}is an(1,**F***ν*) 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

The distribution of the sum of independent random variables is the convolution of their distributions. Suppose is the sum of independent random variables each with probability mass functions . Then

has

If it has a distribution from the same family of distributions as the original variables, that family of distributions is said to be *closed under convolution*.

Examples of such univariate distributions are: normal distributions, Poisson distributions, binomial distributions (with common success probability), negative binomial distributions (with common success probability), gamma distributions (with common rate parameter), chi-squared distributions, Cauchy distributions, hyperexponential distributions.

**Examples: ^{[3]}^{[4]}**

- 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}. - The sum of
**gamma**(*n*_{i},*β*) random variables has a**gamma**(Σ*n*_{i},*β*) distribution. - 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. - 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. - If
*X*_{1}is a**normal**(*μ*_{1},*σ*^{2}_{1}) 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*^{2}_{1}+*σ*^{2}_{2}) random variable. - The sum of
*N*chi-squared (1) random variables has a chi-squared distribution with*N*degrees of freedom.

- If

Other distributions are not closed under convolution, but their sum has a known distribution:

- The sum of
*n***Bernoulli**(p) random variables is a**binomial**(*n*,*p*) 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**(*β*) random variables is a**gamma**(*n*,*β*) 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.

- If the exponential random variables have a common rate parameter, their sum has an
- The sum of the squares of
*N***standard normal**random variables has a**chi-squared**distribution with N degrees of freedom.

### Product of variables

The product of independent random variables *X* and *Y* may belong to the same family of distribution as *X* and *Y*: Bernoulli distribution and log-normal distribution.

**Example: **

- If
*X*_{1}and*X*_{2}are independent**log-normal**random variables with parameters (*μ*_{1},*σ*^{2}_{1}) and (*μ*_{2},*σ*^{2}_{2}) respectively, then*X*_{1}*X*_{2}is a**log-normal**random variable with parameters (*μ*_{1}+*μ*_{2},*σ*^{2}_{1}+*σ*^{2}_{2}).

### Minimum and maximum of independent random variables

For some distributions, the **minimum** value of several independent random variables is a member of the same family, with different parameters:
Bernoulli distribution, Geometric distribution, Exponential distribution, Extreme value distribution, Pareto distribution, Rayleigh distribution, Weibull distribution.

**Examples: **

- If
*X*_{1}and*X*_{2}are independent**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 if expressed in terms probability of failure:*q*=*q*_{1}*q*_{2}. - If
*X*_{1}and*X*_{2}are independent**exponential**random variables with rate*μ*_{1}and*μ*_{2}respectively, then min(*X*_{1},*X*_{2}) is an exponential random variable with rate*μ*=*μ*_{1}+*μ*_{2}.

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.

### Other

- If
*X*and*Y*are independent**standard normal**random variables,*X*/*Y*is a**Cauchy**(0,1) random variable. - If
*X*_{1}and*X*_{2}are independent**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. - 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**ν*) random variable. - If
*X*_{1}is a**gamma**(*α*_{1}, 1) random variable and*X*_{2}is an independent**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 a gamma(*α*_{1},*β*_{1}) random variable and*X*_{2}is an independent 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. - 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

Approximate or limit relationship means

- 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.

**Combination of iid random variables: **

- Given certain conditions, the sum (hence the average) 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 (CLT).

**Special case of distribution parametrization: **

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

Consequences of the CLT:

- If
*X*is a**Poisson**random variable with large mean, then for integers*j*and*k*, P(*j*≤*X*≤*k*) approximately equals to*P*(*j*− 1/2 ≤*Y*≤*k*+ 1/2) where*Y*is a**normal**distribution with the same mean and variance as*X*. - If
*X*is a**binomial**(*n*,*p*) random variable with large*np*and*n*(1 −*p*), then for integers*j*and*k*, P(*j*≤*X*≤*k*) approximately equals to P(*j*− 1/2 ≤*Y*≤*k*+ 1/2) where*Y*is a**normal**random variable with the same mean and variance as*X*, i.e.*np*and*np*(1 −*p*). - 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*/(*α*+*β*) and variance*αβ*/((*α*+*β*)^{2}(*α*+*β*+ 1)). - If
*X*is a**gamma**(*α*,*β*) 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**random variable with a large number of degrees of freedom*t**ν*then*X*approximately has a**standard normal**distribution. - If
*X*is an(**F***ν*,*ω*) 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

When one or more parameter(s) of a distribution are random variables, the compound distribution is the marginal distribution of the variable.

**Examples: **

- If
*X*|*N*is a**binomial**(*N*,*p*) random variable, where parameter*N*is a random variable with negative-binomial (*m*,*r*) distribution, then*X*is distributed as a**negative-binomial**(*m*,*r*/(*p*+*qr*)). - If
*X*|*N*is a**binomial**(*N*,*p*) random variable, where parameter*N*is a random variable with Poisson(*μ*) distribution, then*X*is distributed as a**Poisson**(*μp*). - If
*X*|*μ*is a**Poisson**(*μ*) random variable and parameter*μ*is random variable with gamma(*m*,*θ*) distribution (where*θ*is the scale parameter), then*X*is distributed as a**negative-binomial**(*m*,*θ*/(1 +*θ*)), sometimes called gamma-Poisson distribution.

Some distributions have been specially named as compounds: beta-binomial distribution, beta-Pascal distribution, gamma-normal distribution.

**Examples: **

- If
*X*is a Binomial(*n*,*p*) random variable, and parameter p is a random variable with beta(*α*,*β*) distribution, then*X*is distributed as a Beta-Binomial(*α*,*β*,*n*). - If
*X*is a negative-binomial(*m*,*p*) random variable, and parameter*p*is a random variable with beta(*α*,*β*) distribution, then*X*is distributed as a Beta-Pascal(*α*,*β*,*m*).

## See also

## References

**^**LEEMIS, Lawrence M.; Jacquelyn T. MCQUESTON (February 2008). "Univariate Distribution Relationships" (PDF).*American Statistician*.**62**(1): 45–53. doi:10.1198/000313008x270448.**^**Swat, MJ; Grenon, P; Wimalaratne, S (2016). "ProbOnto: ontology and knowledge base of probability distributions".*Bioinformatics*.**32**(17): 2719–21. doi:10.1093/bioinformatics/btw170. PMC 5013898. PMID 27153608.**^**Cook, John D. "Diagram of distribution relationships".**^**Dinov, Ivo D.; Siegrist, Kyle; Pearl, Dennis; Kalinin, Alex; Christou, Nicolas (2015). "Probability Distributome: a web computational infrastructure for exploring the properties, interrelations, and applications of probability distributions".*Computational Statistics*.**594**(2): 249–271. doi:10.1007/s00180-015-0594-6. PMC 4856044. PMID 27158191.

## External links

- Interactive graphic: Univariate Distribution Relationships
- ProbOnto - Ontology and knowledge base of probability distributions: ProbOnto
- Probability Distributome project includes calculators, simulators, experiments, and navigators for inter-distributional refashions and distribution meta-data.