Page 222 - Elements of Distribution Theory

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052184472Xc07 CUNY148/Severini May 24, 2005 3:59

208 Distribution Theory for Functions of Random Variables

It follows that the marginal density of Y 1 is given by
1 ∞ ν+1 −1 1 2
√ ν y 2 2 exp − y /ν + 1 y 2 dy 2
1
ν
(2πν)2 2 0 2
2
ν+1
1 2 2 −(ν+1)/2
= √ y /ν + 1 , y 2 ∈ R.
1
(πν) ν
2
This is the density of the t-distribution with ν degrees of freedom.
Example 7.11 (F-distribution). Let X 1 and X 2 denote independent random variables such
that X 1 has a chi-squared distribution with ν 1 degrees of freedom and X 2 has a chi-squared
distribution with ν 2 degrees of freedom. Let
X 1 /ν 1
Y 1 = ;
X 2 /ν 2
the distribution of Y 1 is called the F-distribution with (ν 1 ,ν 2 ) degrees of freedom. The
density of this distribution may be determined using Theorem 7.2.
Let Y 2 = X 2 . Writing X = (X 1 , X 2 ) and Y = (Y 1 , Y 2 ), X = h(Y) where h = (h 1 , h 2 ),

ν 1
h 1 (y) = y 1 y 2 , h 2 (y) = y 2 .
ν 2
Hence,

ν 1
∂h(y)
= y 2 .
∂y ν 2

The density of X is given by
1 ν 1 −1 ν 2 −1 1 2
p X (x) = (ν 1 +ν 2 ) x 1 2 x 2 2 exp − (x 1 + x 2 ) , x ∈ (0, ∞) .
2 2 ( ) ( ) 2
ν 1
ν 2
2 2
Hence, by Theorem 7.2, Y has density
ν 1
(ν 1 /ν 2 ) 2 ν 1 −1 ν 1 +ν 2 −1 1 ν 1 2
p Y (y) = y 2 y 2 exp − y 1 + 1 y 2 , y ∈ (0, ∞) .
(ν 1 +ν 2 ) 1 2
2 2 ν 1 ν 2 2 ν 2
2 2
It follows that the marginal density of Y 1 is given by
ν 1
∞
(ν 1 /ν 2 ) 2 ν 1 −1 ν 1 +ν 2 −1 1 ν 1
y 2 y 2 exp −
(ν 1 +ν 2 ) 1 2 y 1 + 1 y 2 dy 2
2 2 ν 1 ν 2 0 2 ν 2
2 2
ν 1 ν 1 −1
ν 1 +ν 2 ν 1 2 y 2
= 2 1 , y 1 ∈ (0, ∞).
ν 1 ν 2 ν 2 ν 1 +ν 2
2 2 ν 1 y 1 + 1 2
ν 2
This is the density of the F-distribution with (ν 1 ,ν 2 )degrees of freedom.
Functions that are not one-to-one
Even in cases in which the dimension of Y is the same as the dimension of X,itisnot
possible to apply Theorem 7.2 if the function g is not one-to-one. However, if the set X
may be partitioned into subsets such that g is one-to-one on each subset, then the change-
of-variable formula may be applied on each subset. The results may then be combined to
obtain the result for g(X).