Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.15 IMPORTANT JOINT DISTRIBUTIONS

where

J≡

∂(x 1 ,x 2 ...,xn)

∂(y 1 ,y 2 ,...,yn)

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∂x 1

∂y 1

...

∂xn

∂y 1

..

.

..

.

..

.

∂x 1

∂yn

...

∂xn

∂yn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

,

is the Jacobian of thexiwith respect to theyj.

Suppose that the random variablesXi,i=1, 2 ,...,n, are independent and Gaussian dis-

tributed with meansμiand variancesσ^2 irespectively. Find the PDF for the new variables

Zi=(Xi−μi)/σi,i=1, 2 ,...,n. By considering an elemental spherical shell inZ-space,

find the PDF of the chi-squared random variableχ^2 n=

∑n

i=1Z

2

i.

Since theXiare independent random variables,

f(x 1 ,x 2 ,...,xn)=f(x 1 )f(x 2 )···f(xn)=

1

(2π)n/^2 σ 1 σ 2 ···σn

exp

[

−

∑n

i=1

(xi−μi)^2

2 σ^2 i

]

.

To derive the PDF for the variablesZi,werequire

|f(x 1 ,x 2 ,...,xn)dx 1 dx 2 ···dxn|=|g(z 1 ,z 2 ,...,zn)dz 1 dz 2 ···dzn|,

and, noting thatdzi=dxi/σi,weobtain

g(z 1 ,z 2 ,...,zn)=

1

(2π)n/^2

exp

(

−

1

2

∑n

i=1

z^2 i

)

.

Let us now consider the random variableχ^2 n=

∑n

i=1Z

2
i, which we may regard as the
square of the distance from the origin in then-dimensionalZ-space. We now require that

g(z 1 ,z 2 ,...,zn)dz 1 dz 2 ···dzn=h(χ^2 n)dχ^2 n.

If we consider the infinitesimal volumedV=dz 1 dz 2 ···dznto be that enclosed by the
n-dimensional spherical shell of radiusχnand thicknessdχnthen we may writedV=
Aχnn−^1 dχn, for some constantA. We thus obtain

h(χn^2 )dχ^2 n ∝ exp(−^12 χ^2 n)χnn−^1 dχn ∝ exp(−^12 χ^2 n)χnn−^2 dχ^2 n,

where we have used the fact thatdχ^2 n=2χndχn. Thus we see that the PDF forχ^2 nis given
by

h(χ^2 n)=Bexp(−^12 χ^2 n)χnn−^2 ,

for some constantB. This constant may be determined from the normalisation condition
∫∞

0

h(χ^2 n)dχ^2 n=1

and is found to beB=[2n/^2 Γ(^12 n)]−^1. This is thenth-order chi-squared distribution
discussed in subsection 30.9.4.

30.15 Important joint distributions

In this section we will examine two important multivariate distributions, the

multinomial distribution, which is an extension of the binomial distribution, and

themultivariate Gaussian distribution.