TransformedDistribution

The distribution of $expr(x)$ where $x \sim dist$ — exact for $expr$ affine in $x$ , plus $x^2$ of a standard normal.

TransformedDistribution(expr, Distributed(x, dist))the distribution of expr(x) for x ~ dist; exact for expr affine in x, plus x^2 of a standard normal.

Typed signatures

Type	Package	Applies
`(any, expression<Distributed>) -> distribution`	`statistics`	always

Domain: Statistics

Details

$expr = a\,x + b$ ( $a \ne 0$ ): change of variables — $PDF(y) = PDF(dist,(y-b)/a)/|a|$ ; $CDF(y) = CDF(dist,(y-b)/a)$ for $a>0$ , or $1 - CDF(dist,(y-b)/a)$ (plus $PDF(dist,(y-b)/a)$ when $dist$ is discrete, since the inequality flips through an equality) for $a<0$ ; $Mean = a\,Mean(dist)+b$ ; $Variance = a^2\,Variance(dist)$ exactly.
$expr = x^2$ , $dist = $ NormalDistribution $(0,1)$ : ChiSquareDistribution(1) by definition — PDF/CDF/Mean/Variance/RandomVariate all forward to it.
$expr = Norm(\{x_1,\dots,x_k\})$ over $k$ independent standard normals: ChiDistribution $(k)$ , which is RayleighDistribution $(1)$ for $k=2$ and HalfNormalDistribution $(\sqrt{\pi/2})$ for $k=1$ .
Any other shape of $expr$ , or a non-affine one with a non-standard-normal $x^2$ base, stays unevaluated — no numeric approximation.

Examples

Basic #

In

\mathrm{mean}(\mathrm{TransformedDistribution}(2x+3, \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1))))

Out

3

In

\mathrm{Variance}(\mathrm{TransformedDistribution}(2x+3, \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1))))

Out

4

In

\operatorname{CDF}(\mathrm{TransformedDistribution}(-x, \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1))), 0)

Out

\frac{1}{2}

In

\mathrm{mean}(\mathrm{TransformedDistribution}(x^2, \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1))))

Out

1

Properties #

In

\mathrm{TransformedDistribution}(\left\Vert \bigl\lbrack x, y, w, z\bigr\rbrack\right\Vert, \mathrm{Distributed}(\bigl\lbrack x, y, w, z\bigr\rbrack, \mathrm{ProductDistribution}(\bigl\lbrack\operatorname{NormalDistribution}(), 4\bigr\rbrack)))

Out

\mathrm{ChiDistribution}(4)

In

\mathrm{TransformedDistribution}(\left\Vert \bigl\lbrack x, y\bigr\rbrack\right\Vert, \mathrm{Distributed}(\bigl\lbrack x, y\bigr\rbrack, \mathrm{ProductDistribution}(\bigl\lbrack\operatorname{NormalDistribution}(), 2\bigr\rbrack)))

Out

\mathrm{RayleighDistribution}(1)

In

\mathrm{TransformedDistribution}(\left\Vert \bigl\lbrack x\bigr\rbrack\right\Vert, \mathrm{Distributed}(\bigl\lbrack x\bigr\rbrack, \mathrm{ProductDistribution}(\bigl\lbrack\operatorname{NormalDistribution}(), 1\bigr\rbrack)))

Out

\mathrm{HalfNormalDistribution}(\sqrt{\frac{π}{2}})

The ratio of two independent variables is a transformed distribution, simplified to $u/v$

In

\mathrm{TransformedDistribution}(\frac{u}{v}, \mathrm{Distributed}(\bigl\lbrack u, v\bigr\rbrack, \mathrm{ProductDistribution}(\bigl\lbrack\operatorname{NormalDistribution}(), 2\bigr\rbrack)))

Out

\mathrm{CauchyDistribution}(0, 1)

About

Examples	8
Checked against	Wolfram Language 4 agree

See also: Distributed, ChiSquareDistribution, Mean, Variance