3.6.t-andF-Distributions 211

Define a new random variableTby writing

T=

W

√

V/r

. (3.6.1)

The transformation technique is used to obtain the pdfg 1 (t)ofT. The equations

t=

w

√

v/r

and u=v

define a transformation that mapsS ={(w, v):−∞<w<∞, 0 <v<∞}
one-to-one and ontoT ={(t, u):−∞<t<∞, 0 <u<∞}.Sincew =
t

√

u/

√
√ r, v=u, the absolute value of the Jacobian of the transformation is|J|=
u/

√

r. Accordingly, the joint pdf ofTandU=V is given by

g(t, u)=h

(

t

√

u

√

r

,u

)

|J|

=

{

√^1

2 πΓ(r/2)2r/^2 u

r/ 2 − (^1) exp
[
−u 2
(
1+t
2
r
)]√
√u
r |t|<∞,^0 <u<∞
0elsewhere.
The marginal pdf ofTis then
g 1 (t)=
∫∞
−∞
g(t, u)du

∫∞
0
1
√
2 πrΓ(r/2)2r/^2
u(r+1)/^2 −^1 exp
[
−
u
2
(
1+
t^2
r
)]
du.
In this integral letz=u[1 + (t^2 /r)]/2, and it is seen that
g 1 (t)=
∫∞
0
1
√
2 πrΓ(r/2)2r/^2
(
2 z
1+t^2 /r
)(r+1)/ 2 − 1
e−z
(
2
1+t^2 /r
)
dz

Γ[(r+1)/2]
√
πrΓ(r/2)
1
(1 +t^2 /r)(r+1)/^2
, −∞<t<∞. (3.6.2)
Thus, ifWisN(0,1),Visχ^2 (r), andWandVare independent, thenT=W/
√
V/r
has the pdfg 1 (t), (3.6.2). The distribution of the random variableT is usually
called at-distribution. It should be observed that at-distribution is completely
determined by the parameterr, the number of degrees of freedom of the random
variable that has the chi-square distribution.
The pdfg 1 (t)satisfiesg 1 (−t)=g 1 (t); hence, the pdf ofTis symmetric about 0.
Thus, the median ofTis 0. Upon differentiatingg 1 (t), it follows that the unique
maximum of the pdf occurs at 0 and that the derivative is continuous. So, the pdf is
mound shaped. As the degrees of freedom approach∞,thet-distribution converges
to theN(0,1) distribution; see Example 5.2.3 of Chapter 5.
The R commandpt(t,r)computes the probabilityP(T≤t)whenT has a
t-distribution withrdegrees of freedom. For instance, the probability that at-
distributed random variable with 15 degrees of freedom is less than 2.0 is computed