7.37:
a. W follows a chi-squared distribution with 5 degrees of freedom. See theorem 7.2 from the same chapter, which says
[tex]\displaystyle \sum_{i=1}^n\left(\frac{Y_i-\mu}{\sigma}\right)^2[/tex]
is chi-squared distributed with n d.f.. Here we have [tex]\mu=0[/tex] and [tex]\sigma=1[/tex].
b. U follows a chi-squared distribution with 4 degrees of freedom. See theorem 7.3:
[tex]\displaystyle \frac1{\sigma^2}\sum_{i=1}^n (Y_i-\overline Y)^2[/tex]
is chi-squared distributed with n - 1 d.f..
c. Y₆² is chi-square distributed for the same reason as W, but with d.f. = 1. The sum of chi-squared distributed random variables is itself chi-squared distributed, with d.f. equal to the sum of the individual random variables' d.f.s. Then U + Y₆² is chi-squared distributed with 5 + 1 = 6 degrees of freedom.
7.38:
a. Notice that
[tex]\dfrac{\sqrt 5 Y_6}{\sqrt W} = \dfrac{Y_6}{\sqrt{\frac W5}}[/tex]
and see definition 7.2 for the t distribution. Since Y₆ is normally distributed with mean 0 and s.d. 1, it follows that this random variable is t distributed with 5 degrees of freedom.
b. Similar manipulation gives
[tex]\dfrac{2Y_6}{\sqrt U} = \dfrac{\sqrt4 Y_6}{\sqrt U} = \dfrac{Y_6}{\sqrt{\frac U4}}[/tex]
so this r.v. is t distributed with 4 degrees of freedom.
