a. Let X1 have a chi-squared distribution with parameter 1

(see Section 4.4), and let X2 be independent of X1 and have a chi-squared distribution with parameter 2. Use the technique of Example 5.21 to show that X1 X2 has a chi-squared distribution with parameter 1 2.

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
1. Let Z1, Z2, . . . , Zn be n independent standard normal rv’s. What is the distribution of Z2 1 . . . Z2 n? Justify your answer.

c. Let X1, . . . , Xn be a random sample from a normal distribution with mean and variance 2

. What is the distribution of the sum Y

n i
1 [(Xi  )/]2

? Justify your answer.