a. The event { } is equivalent to what event involving X itself?

b. If X has a standard normal distribution, use part

(a) to write the integral that equals . Then differentiate this with respect to y to obtain the pdf of [the square of a N(0, 1) variable]. Finally, show that has a chi-squared distribution with df [see (4.10)].

[Hint: Use the following identity.]

d dy e 3 b(y)

a(y)

f(x) dx f 5 f [b(y)] # br

(y) 2 f [a(y)] # ar

(y)