Let U and V be independent chi-square variables, with U ~ ? 2 k and V ~ ? 2 n . Define X = U/k V/n..

Let U and V be independent chi-square variables, with U ∼ χ

2

k

and V ∼ χ

2

n

. Define X =

U/k

V/n

(a) Letting Y = V, obtain the joint density of (X,Y). (b) Show that X has the so-called “Fdistribution with k and n degrees of freedom” with density given by

fX (x) = Γ

k+n

2

(k/n)

k/2

Γ(k/2)Γ(n/2)

x

k

2 −1

1+

kx

n

k+n

2

I(0,∞)(x).

(Note: The F distribution with integer-valued parameters k and n called the distribution’s

“degrees of freedom” was named after Sir Ronald Fisher, one of the true geniuses of the

20th century, and the statistician who laid the foundations for the “analysis of variance”

(ANOVA) where F tests are essential tools for statistical inference.)