Let X u...,Xn be independently distributed random variables and let Sm = Sjij X h m < n. Show that the joint distribution of X t and Sm does not depend on i < m.

6 . Let X u...,Xn be independently and identically distributed normal random variables with E(X]) = |xt, var(AT/) = 1 for i = 1 Let 8 2 =

|x-. Show that the pdf of U = 2f=1 X] is given by f v(i) =

2f=i Pv(i)f(x2n+2i), where p v(i) is the pdf of a Poisson random variable with parameter \ S2, f(xl+n) is the pdf of a central chi-square with n + 2i degrees of freedom and V is independent of xl+v- {Note: This is also known as the pdf of a noncentral chi-square random variable with n degrees of freedom and noncentrality parameter 8 2.)