Mathematical Methods for Physics and Engineering - Matematica.NET

30.15 IMPORTANT JOINT DISTRIBUTIONSwhere∣ ∣∣∣∣∣∣∣∣∣ ∂x 1 ∂x∣n ∣∣∣∣∣∣∣∣∣...J ≡ ∂(x ∂y 1 ∂y 11,x 2 ...,x n )∂(y 1 ,y 2 ,...,y n ) = .. .. . ,∂x 1 ∂x n...∂y n ∂y nis the Jacobian of the x i with respect to the y j .◮Suppose that the random variables X i , i =1, 2,...,n, are independent and Gaussian distributedwith means µ i and variances σi2 respectively. Find the PDF for the new variablesZ i =(X i − µ i )/σ i , i =1, 2,...,n. By considering an elemental spherical shell in Z-space,find the PDF of the chi-squared random variable χ 2 n = ∑ ni=1 Z i 2 .Since the X i are independent random variables,[]1n∑ (x i − µ i ) 2f(x 1 ,x 2 ,...,x n )=f(x 1 )f(x 2 ) ···f(x n )=exp −.(2π) n/2 σ 1 σ 2 ···σ n 2σ 2 i=1 iTo derive the PDF for the variables Z i ,werequire|f(x 1 ,x 2 ,...,x n ) dx 1 dx 2 ···dx n | = |g(z 1 ,z 2 ,...,z n ) dz 1 dz 2 ···dz n |,and, noting that dz i = dx i /σ i ,weobtain)1g(z 1 ,z 2 ,...,z n )=(−(2π) exp 1 n∑z n/2 i2 .2i=1Let us now consider the random variable χ 2 n = ∑ ni=1 Z i 2 , which we may regard as thesquare of the distance from the origin in the n-dimensional Z-space. We now require thatg(z 1 ,z 2 ,...,z n ) dz 1 dz 2 ···dz n = h(χ 2 n)dχ 2 n.If we consider the infinitesimal volume dV = dz 1 dz 2 ···dz n to be that enclosed by then-dimensional spherical shell of radius χ n and thickness dχ n then we may write dV =dχ n , for some constant A. We thus obtainAχ n−1nh(χ 2 n)dχ 2 n ∝ exp(− 1 2 χ2 n)χ n−1n dχ n ∝ exp(− 1 2 χ2 n)χn n−2 dχ 2 n,where we have used the fact that dχ 2 n =2χ n dχ n . Thus we see that the PDF for χ 2 n is givenbyh(χ 2 n)=B exp(− 1 2 χ2 n)χ n−2n ,for some constant B. This constant may be determined from the normalisation condition∫ ∞h(χ 2 n) dχ 2 n =10and is found to be B = [2 n/2 Γ( 1 2 n)]−1 . This is the nth-order chi-squared distributiondiscussed in subsection 30.9.4. ◭30.15 Important joint distributionsIn this section we will examine two important multivariate distributions, themultinomial distribution, which is an extension of the binomial distribution, andthe multivariate Gaussian distribution.1207