a. Let X1 have a chi-squared distribution with parameter n1 (see Section 4.4), and let X2 be independent of X1 and have a chi-squared distribution with parameter n2. Use the technique of Example 5.21 to show that X1 X2 has a chi-squared distribution with parameter n1 n2.

f(x, y) 5 •

2 5

(2x 1 3y) 0 # x # 1, 0 # y # 1 0 otherwise

b. In Exercise 71 of Chapter 4, you were asked to show that if Z is a standard normal rv, then Z2 has a chi-squared distribution with n 1.

Let Z1, Z 2, . . . , Zn be n independent standard normal rv’s. What is the distribution of

? Justify your answer.

c. Let X1, . . . , Xn be a random sample from a normal distribution with mean m and variance s2

. What is the distribution of the sum ? Justify your answer.