71.

a. The event {X2  y} 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 P(X2  y). Then differentiate this with respect to y to obtain the pdf of X2 [the square of a N(0, 1) variable]. Finally, show that X2 has a chi-squared distribution with
1 df [see (4.10)].

[Hint: Use the following identity.]