Chapter 4.4, Problem 71E

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 ofX² [the square of a N(0, 1) variable]. Finally, show that X² has a chi-squared distribution with v = 1 df [see (4.10)]. [Hint: Use the following identity.]

     $\frac{d}{dy}\left\{{\displaystyle {\int }_{a(y)}^{b(y)}f(x)dx}\right\}=f[b(y)].b^{\prime }(y)-f[a(y)].a^{\prime }(y)$