3.6.t-andF-Distributions 213

We define the new random variable

W=

U/r 1

V/r 2

and we propose finding the pdfg 1 (w)ofW. The equations

w=

u/r 1

v/r 2

,z=v,

define a one-to-one transformation that maps the setS={(u, v):0<u<∞, 0 <
v<∞}onto the setT ={(w, z):0<w<∞, 0 <z<∞}.Sinceu=
(r 1 /r 2 )zw, v=z, the absolute value of the Jacobian of the transformation is
|J|=(r 1 /r 2 )z. The joint pdfg(w, z) of the random variablesWandZ=Vis then

g(w, z)=

1

Γ(r 1 /2)Γ(r 2 /2)2(r^1 +r^2 )/^2

(

r 1 zw

r 2

)r 12 −^2

z

r 2 − 2

(^2) exp
[
−
z
2
(
r 1 w
r 2
+1
)]
r 1 z
r 2
,
provided that (w, z)∈T, and zero elsewhere. The marginal pdfg 1 (w)ofWis then
g 1 (w)=
∫∞
−∞
g(w, z)dz

∫∞
0
(r 1 /r 2 )r^1 /^2 (w)r^1 /^2 −^1
Γ(r 1 /2)Γ(r 2 /2)2(r^1 +r^2 )/^2
z(r^1 +r^2 )/^2 −^1 exp
[
−
z
2
(
r 1 w
r 2
+1
)]
dz.
If we change the variable of integration by writing
y=
z
2
(
r 1 w
r 2
+1
)
,
it can be seen that
g 1 (w)=
∫∞
0
(r 1 /r 2 )r^1 /^2 (w)r^1 /^2 −^1
Γ(r 1 /2)Γ(r 2 /2)2(r^1 +r^2 )/^2
(
2 y
r 1 w/r 2 +1
)(r 1 +r 2 )/ 2 − 1
e−y
×
(
2
r 1 w/r 2 +1
)
dy

{
Γ[(r 1 +r 2 )/2](r 1 /r 2 )r^1 /^2
Γ(r 1 /2)Γ(r 2 /2)
wr^1 /^2 −^1
(1+r 1 w/r 2 )(r 1 +r 2 )/^20 <w<∞
0elsewhere.
(3.6.6)
Accordingly, ifUandV are independent chi-square variables withr 1 andr 2
degrees of freedom, respectively, thenW=(U/r 1 )/(V/r 2 )hasthepdfg 1 (w), (3.6.6).
The distribution of this random variable is usually called anF-distribution;and
we often call the ratio, which we have denoted byW, F.Thatis,
F=
U/r 1
V/r 2

. (3.6.7)

It should be observed that anF-distribution is completely determined by the two

parametersr 1 andr 2.