15. Let X and Y be independent random variables, X having the normal distribution with mean 0 and variance 1, and Y having the 2

15. Let X and Y be independent random variables, X having the normal distribution with mean 0 and variance 1, and Y having the χ2 distribution with n degrees of freedom. Show that T =

X

√Y/n has density function f (t ) =

1

√πn

Ŵ( 1 2 (n + 1))

Ŵ( 1 2n)

1 +

t2 n

!

−1 2 (n+1)

for t ∈ R.

T is said to have the t -distribution with n degrees of freedom.

16. Let X and Y be independent random variables with the χ2 distribution, X having m degrees of freedom and Y having n degrees of freedom. Show that U =

X/m Y/n has density function f (u) =

mŴ( 1 2 (m + n))

nŴ( 1 2m)Ŵ( 1 2 n) ·

(mu/n)

1 2m−1



1 + (mu/n)

 1 2 (m+n)

for u > 0.

U is said to have the F-distribution with m and n degrees of freedom.