Question

A random variable that has an mgf of the functional form M(t) = frac{1}{(1-2t)^{r/2}} exp left( frac{t heta}{1-2t} ight) where t < 1/2, heta geq 0, and r is a positive integer, is said to have a noncentral chi-square distribution with r degrees of freedom and noncentral parameter heta, written as chi^2(r, heta). (a) Let X sim chi^2(r, heta). Calculate E(X) and Var(X). (b) Let X_1, dots, X_n denote independent random variables that are N(mu_i, sigma^2), i = 1, dots, n. Let Y = sum_{i=1}^n X_i^2 / sigma^2. Show that the mgf of Y is given by M_Y(t) = frac{1}{(1-2t)^{n/2}} exp left{ frac{t sum_{i=1}^n mu_i^2}{sigma^2(1-2t)} ight}, t < frac{1}{2} That is, Y sim chi^2(n, sum_{i=1}^n mu_i^2 / sigma^2). (c) Let X_1, dots, X_n be independent random variables that are chi^2(r_i, heta_i), i = 1, dots, n. Let Y = sum_{i=1}^n X_i. Show that Y sim chi^2(sum_{i=1}^n r_i, sum_{i=1}^n heta_i).

          A random variable that has an mgf of the functional form

M(t) = frac{1}{(1-2t)^{r/2}} exp left( frac{t 	heta}{1-2t} 
ight)

where t < 1/2, 	heta geq 0, and r is a positive integer, is said to have a noncentral chi-square distribution with r degrees of freedom and noncentral parameter 	heta, written as chi^2(r, 	heta).

(a) Let X sim chi^2(r, 	heta). Calculate E(X) and Var(X).

(b) Let X_1, dots, X_n denote independent random variables that are N(mu_i, sigma^2), i = 1, dots, n. Let Y = sum_{i=1}^n X_i^2 / sigma^2. Show that the mgf of Y is given by

M_Y(t) = frac{1}{(1-2t)^{n/2}} exp left{ frac{t sum_{i=1}^n mu_i^2}{sigma^2(1-2t)} 
ight}, t < frac{1}{2}

That is, Y sim chi^2(n, sum_{i=1}^n mu_i^2 / sigma^2).

(c) Let X_1, dots, X_n be independent random variables that are chi^2(r_i, 	heta_i), i = 1, dots, n. Let Y = sum_{i=1}^n X_i. Show that Y sim chi^2(sum_{i=1}^n r_i, sum_{i=1}^n 	heta_i).

$A random variable that has an mgf of the functional form M(t) = frac1(1-2t)^r/2 exp left( fract heta1-2t ight) where t < 1/2, heta geq 0, and r is a positive integer, is said to have a noncentral chi-square distribution with r degrees of freedom and noncentral parameter heta, written as chi^2(r, heta). (a) Let X sim chi^2(r, heta). Calculate E(X) and Var(X). (b) Let X1, dots, Xn denote independent random variables that are N(mui, sigma^2), i = 1, dots, n. Let Y = sumi=1^n Xi^2 / sigma^2. Show that the mgf of Y is given by MY(t) = frac1(1-2t)^n/2 exp left fract sumi=1^n mui^2sigma^2(1-2t) ight, t < frac12 That is, Y sim chi^2(n, sumi=1^n mui^2 / sigma^2). (c) Let X1, dots, Xn be independent random variables that are chi^2(ri, hetai), i = 1, dots, n. Let Y = sumi=1^n Xi. Show that Y sim chi^2(sumi=1^n ri, sumi=1^n hetai).$

Added by Susan R.

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