Page 218 - Elements of Distribution Theory

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

204 Distribution Theory for Functions of Random Variables

The inverse of this function is given by h = (h 1 ,..., h n ) with

h j (y) = y j /y j−1 , j = 1, 2,..., n;

here y = (y 1 ,..., y n ) and y 0 = 1.
The density function of X is given by

n
p X (x) = 1, x ∈ (0, 1) .
n
We may take X 0 = (0, 1) ; then
n
g(X 0 ) ={y = (y 1 ,..., y n ) ∈ R :0 < y n < y n−1 < ··· < y 1 < 1}.
It is easy to see that the matrix ∂h(y)/∂y is a lower triangular matrix with diagonal elements
1, 1/y 1 ,..., 1/y n−1 . Hence,
1

∂h(y)
= .
∂y y 1 ··· y n−1

It follows that Y has an absolutely continuous distribution with density function
1
p Y (y) = , 0 < y n < y n−1 < ··· < y 1 < 1.
y 1 ··· y n−1

Functions of lower dimension
d
Let X denote a random variable with range X ⊂ R , d ≥ 2. Suppose we are interested
q
in the distribution of g 0 (X) where g 0 : X → R , q < d. Note that, since the dimension
of g 0 (X)is less than the dimension of X, Theorem 7.2 cannot be applied directly. To use
Theorem 7.2 in these cases, we can construct a function g 1 such that g = (g 0 , g 1 ) satisfies
the conditions of Theorem 7.2. We may then use Theorem 7.2 to find the density of g(X) and
then marginalize to find the density of g 0 (X). This approach is illustrated on the following
examples.

Example 7.7 (Ratios of exponential random variables to their sum). Let X 1 , X 2 ,..., X n
denote independent, identically distributed random variables, each with a standard expo-
nential distribution. Let
X 1 X 2 X n−1
Y 1 = , Y 2 = ,. .., Y n−1 =
X 1 +· · · + X n X 1 +· · · + X n X 1 +· · · + X n
and suppose we want to find the distribution of the random vector (Y 1 ,..., Y n−1 ).
Clearly, the function mapping (X 1 ,..., X n )to(Y 1 ,..., Y n−1 )is not one-to-one. Let Y n =
X 1 +· · · + X n . Then, writing Y = (Y 1 ,..., Y n ) and X = (X 1 ,..., X n ), we have Y = g(X)
where g = (g 1 ,..., g n )isgiven by
x j
g n (x) = x 1 + ··· + x n , g j (x) = , j = 1,..., n − 1.
g n (x)
The function g is one-to-one, with inverse h = (h 1 ,..., h n ) where

h j (y) = y j y n , j = 1,..., n − 1, and h n (y) = (1 − y 1 −· · · − y n−1 )y n .