Dab solver - Relationships among probability distributions

Most tools require the use of JavaScript. If you are using Firefox with the NoScript addon, please mark toolserver.org as trusted. See the Help page for details.

Feedback / Report a bug

2 points on this page / 19,509 points earned collectively this month / Disambiguate pages on your watchlist

* A [[chi-squared distribution]] with 2 degrees of freedom is an [[exponential distribution]] with mean 2 and vice versa.

* A [[Weibull distribution|Weibull]] (1, β) random variable is an [[exponential distribution|exponential]] random variable with mean β.

* A [[beta distribution|beta]] random variable with parameters α = β = 1 is a  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  with one degree of freedom is a [[Cauchy distribution|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 distribution]], [[Rayleigh distribution]].

Example: 
* If X is a gamma random variable with parameters (r, &lambda;), then Y=aX is a gamma random variable with parameters (r, a&lambda;). 
90 lines hidden (6940 characters)
===  Other  ===

* If X and Y are independent standard normal random variables, X/Y is a Cauchy (0,1) random variable.

* 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.

* If X is a standard normal random variable and U is a chi-squared random variable with ν degrees of freedom, then <math>\frac{X}{\sqrt{(U/ν)}} </math>is a Student's t (ν) random variable.

* 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 X₁is 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.

* If X and Y are exponential random variables with mean μ, then X-Y is a  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,

Single scrollbar

In [[probability theory]] and [[statistics]], there are several relationships among [[probability distributions]]. These relations can be categorized in five 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).

[[File:Relationships among some of univariate probability distributions.jpg|thumb|700px|center|Relationships among some of univariate probability distributions are illustrated with connected lines. dashed lines means approximate relationship. more info:<ref>{{cite journal|last=LEEMIS|first=Lawrence M.|coauthors=Jacquelyn T. MCQUESTON|title=Univariate Distribution Relationships|journal=American Statistician|year=2008|month=February|volume=62|issue=1|pages=45–53|url=http://www.math.wm.edu/~leemis/2008amstat.pdf}}</ref>]]

==Special case of distribution parametrization==

* A [[binomial distribution|binomial]] (n, p) random variable with n  = 1, is a [[Bernoulli distribution|Bernoulli]] (p) random variable.

* A [[negative binomial distribution]] with r = 1 is a [[geometric distribution]].

* A [[gamma distribution]] with shape parameter α = 1 and scale parameter β is an [[exponential distribution|exponential]] (β) distribution.

* A [[gamma distribution|gamma]] (α, β) random variable with α = ν/2 and β = 2, is a [[chi-squared distribution|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 distribution|Weibull]] (1, β) random variable is an [[exponential distribution|exponential]] random variable with mean β.

* A [[beta distribution|beta]] random variable with parameters α = β = 1 is a <<link:0>> random variable.

* A [[beta-binomial distribution|beta-binomial]] (n, 1, 1) random variable is a discrete uniform random variable over the values 0 ... n.

* A random variable with a <<link:1>> with one degree of freedom is a [[Cauchy distribution|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 distribution]], [[Rayleigh distribution]].

'''Example: '''
* If X is a gamma random variable with parameters (r, &lambda;), then Y=aX is a gamma random variable with parameters (r, a&lambda;).

=== Linear function of a random variable ===

The affine transfom ''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, σ2=s2), then X=aZ+b is a normal random variable with parameters (μ=am+b, σ2=a2s2).

=== 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 = μ2 + σ2.
* If X is an F(ν1, ν2) random variable then 1/X is an F(ν2, ν1) 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 ''F''''X''(''X'') is a standard '''uniform''' (0,1) random variable

* If ''X'' is a '''normal''' (μ, σ2) random variable then ''e''''X'' 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.

* 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-distribution|Student’s t]]''' random variable with ν degree of freedom, then ''X''2 is an '''''F''''' (1,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 and ceiling functions|floor]] of an '''exponential''' random variable.

* A '''[[rectangular distribution|rectangular]]''' random variable is the floor of a '''uniform''' random variable.

== Functions of several variables ==

=== Sum of variables ===
The distribution of the sum of independent random variables is called the [[Convolution of probability distributions|convolution]] of the primal distribution.

*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 distribution]]s are:
[[Normal distribution]], [[Poisson distribution]], [[Binomial distribution]] (with common success probability), , [[Negative binomial distribution]] (with common success probability), [[Gamma distribution]](with common rate parameter), [[Chi-squared distribution]], [[Cauchy distribution]], [[Hyper-exponential distribution]].

'''Examples:<ref>{{cite web|last=Cook|first=John D.|title=Diagram of distribution relationships|url=http://www.johndcook.com/distribution_chart.html}}</ref>'''{{citation needed|date=January 2013|reason=no proofs in Cook ref}}
**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''' (''n''i , β) random variables has a '''gamma '''(&Sigma;''n''i , β) distribution.
**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.
**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''' (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.
**The sum of N chi-squared (1) random variables has a chi-squared distribution with N degrees of freedom.

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.

*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 X1 and X2 are independent '''log-normal''' random variables with parameters (μ1, σ12) and (μ2, σ22) respectively, then X1 X2 is a '''log-normal''' random variable with parameters (μ1 + μ2, σ12 + σ22).

=== 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 X1 and X2 are independent '''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 if expressed in terms probability of failure: q = q1 q2.
* If X1 and X2 are independent '''exponential''' random variables with mean μ1 and μ2 respectively, then min(X1, X2) is an exponential random variable with mean μ1 μ2/(μ1 + μ2).

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 distribution]].

=== Other ===

* If ''X'' and ''Y'' are independent '''standard normal''' random variables, ''X''/''Y'' is a '''Cauchy''' (0,1) random variable.

* 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.

* If X is a '''standard normal''' random variable and U is a '''chi-squared''' random variable with ν degrees of freedom, then <math>\frac{X}{\sqrt{(U/ν)}} </math>is a '''Student's ''t''''' (ν) random variable.

* 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 X1is 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.

* If ''X'' and ''Y'' are '''exponential''' random variables with mean μ, then ''X''-''Y'' is a '''<<link:2>>''' 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 [[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 n and np, 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 ''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'''''(ν, ω) random variable with ω large, then ν X 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 probability distribution|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, β) distribution, then X is distributed as a '''negative-binomial''' (m, μβ/(μ+β)), sometimes called [[Gamma-Poisson distribution]] if ''m'' is not integer.

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==

*[[Central limit theorem]]
*[[Compound probability distribution]]

== References ==

== External links ==

* Interactive graphic: [http://www.math.wm.edu/~leemis/chart/UDR/UDR.html Univariate Distribution Relationships]

[[Category:Probability distributions]]

Edit summary:
This is a minor edit Watch this page

Cancel

Dab solver - Relationships among probability distributions

Tools

Interaction