Page 224 - Elements of Distribution Theory

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210 Distribution Theory for Functions of Random Variables

4 + 2 − 2
The set g(∪ X i )is(R ) ∪ (R ) .Itisworth noting that the partition (X 1 ∪ X 2 ), (X 3 ∪
i=1
X 4 ) could also be used, although this choice introduces some minor complications.
The density of X is given by
1 1 2 2 2
p X (x) = exp − x + x 2 , x ∈ R .
1
2π 2
2
2
Consider the transformation on X 1 . Since x + x = y 1 y 2 + y 1 /y 2 , the contribution to the
1 2
density of Y from X 1 is
1 1 + 2
exp − (y 1 y 2 + y 1 /y 2 ) , y ∈ (R ) .
4π|y 2 | 2
It is easy to see that the same result holds for X 4 .
The contribution to the density from either X 2 or X 3 is the same, except that y is restricted
− 2
to (R ) . Hence, the density function of Y is given by
1 1 1 1
− 2
exp − (y 1 y 2 + y 1 /y 2 ) I {y∈(R ) } + exp − (y 1 y 2 + y 1 /y 2 ) I {y∈(R ) }
+ 2
2π|y 2 | 2 2π|y 2 | 2
1 1 + 2 − 2
= exp − (y 1 y 2 + y 1 /y 2 ) , y ∈ (R ) ∪ (R ) .
2π|y 2 | 2
Application of invariance and equivariance
When the distribution of X belongs to a parametric model, it is often convenient to take
advantage of invariance or equivariance when determining the distribution of Y.
Let X denote a random variable with range X and suppose that the distribution of X is
an element of

P ={P(·; θ): θ ∈ }
and that P is invariant with respect to some group of transformations. If Y is a function
of X, and is an invariant statistic, then the distribution of Y does not depend on θ; hence,
when determining the distribution of Y,we may assume that X is distributed according to
P X (·; θ 0 ) where θ 0 is any convenient element of . The resulting distribution for Y does not
depend on the value chosen.

Example 7.13 (Ratios of exponential random variables to their sum). Let X 1 , X 2 ,...,
X n denote independent, identically distributed random variables, each with an exponential
distribution with parameter θ, θ> 0. As in Example 7.8, 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
Recall that the set of exponential distributions with parameter θ ∈ (0, ∞) forms a trans-
formation model with respect to the group of scale transformations; see Example 5.27. Note
that the statistic (Y 1 ,..., Y n−1 )isinvariant under scale transformations: multiplying each
X j by a constant does not change the value of (Y 1 ,..., Y n−1 ). Hence, to determine the
distribution of (Y 1 ,..., Y n−1 )we may assume that X 1 ,..., X n are distributed according to
a standard exponential distribution.
It follows that the distribution of (Y 1 ,..., Y n−1 )is the same as that given in
Example 7.7.