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7.37:
a. <em>W</em> follows a chi-squared distribution with 5 degrees of freedom. See theorem 7.2 from the same chapter, which says
is chi-squared distributed with <em>n</em> d.f.. Here we have and
.
b. <em>U</em> follows a chi-squared distribution with 4 degrees of freedom. See theorem 7.3:
is chi-squared distributed with <em>n</em> - 1 d.f..
c. <em>Y₆</em>² is chi-square distributed for the same reason as <em>W</em>, 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 <em>U</em> + <em>Y₆</em>² is chi-squared distributed with 5 + 1 = 6 degrees of freedom.
7.38:
a. Notice that
and see definition 7.2 for the <em>t</em> distribution. Since <em>Y₆</em> is normally distributed with mean 0 and s.d. 1, it follows that this random variable is <em>t</em> distributed with 5 degrees of freedom.
b. Similar manipulation gives
so this r.v. is <em>t</em> distributed with 4 degrees of freedom.