39.2. THE T AND F DISTRIBUTIONS 805

39.2 The T and F DistributionsThese are really interesting. They both involve independent random variables which aredistributed as normal or X 2 distributions. These involve combinations of these otherrandom variables and the idea is to find the density of these combinations. It is a niceapplication of the change of variables theorem.

39.2.1 The T DistributionHere there are two independent random variables, W which is normally distributed withmean 0 and variance 1 and V which is X 2 (r) . Thus, as explained above,

P((V,W ) ∈ A) =∫

A

1√2π

e−w22

1Γ(r/2)2r/2 v(r/2)−1e−v/2dwdv

thus (V,W ) ∈ (0,∞)× (−∞,∞). The idea is to find the probability density of the statistic

T =W√V/r

It is a random variable which has a known distribution. This involves changing the variable.Let

t =w√v/r

,u = v,

(ut

)= r

(vw

)This maps (0,∞)× (−∞,∞) one to one onto (0,∞)× (−∞,∞) as can be seen with a shortcomputation. Let the density function of (t,u) be f (t,u).

P((t,u) ∈U) = P((v,w) ∈ r−1 (U)

)By the change of variables formula for multiple integrals if U is some open set in R2,∫

Uf (t,u)dudt =

∫r−1(U)

1√2π

e−w22

1Γ(r/2)2r/2 v(r/2)−1e−v/2dwdu

=∫r−1(U)

f

(w√v/r

,v

)J (v,w)dwdu

where

J (v,w) =

∣∣∣∣∣∣det

 1 0− 1

2rw

( 1r v)

32

1√1r v

∣∣∣∣∣∣= 1√1r v

Thus

f

(w√v/r

,v

)1√1r v

=1√2π

e−w22

1Γ(r/2)2r/2 v(r/2)−1e−v/2

Then

f

(w√v/r

,v

)=

1√2π

e−w22

1Γ(r/2)2r/2

√1r

vv(r/2)−1e−v/2