# Dab solver - Relationships among probability distributions

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* 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 $\frac{X}{\sqrt{(U/ν)}}$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,
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