Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

100 Multivariate Distributions

2.2 Transformations:BivariateRandomVariables.............

Let (X 1 ,X 2 ) be a random vector. Suppose we know the joint distribution of
(X 1 ,X 2 ) and we seek the distribution of a transformation of (X 1 ,X 2 ), say,Y =
g(X 1 ,X 2 ). We may be able to obtain the cdf ofY. Another way is to use a trans-
formation as we did for univariate random variables in Sections 1.6 and 1.7. In this
section, we extend this theory to random vectors. It is best to discuss the discrete
and continuous cases separately. We begin with the discrete case.
There are no essential difficulties involved in a problem like the following. Let
pX 1 ,X 2 (x 1 ,x 2 ) be the joint pmf of two discrete-type random variablesX 1 andX 2
withSthe (two-dimensional) set of points at whichpX 1 ,X 2 (x 1 ,x 2 )>0; i.e.,Sis the
support of (X 1 ,X 2 ). Lety 1 =u 1 (x 1 ,x 2 )andy 2 =u 2 (x 1 ,x 2 ) define a one-to-one
transformation that mapsSontoT. The joint pmf of the two new random variables
Y 1 =u 1 (X 1 ,X 2 )andY 2 =u 2 (X 1 ,X 2 )isgivenby

pY 1 ,Y 2 (y 1 ,y 2 )=

{

pX 1 ,X 2 [w 1 (y 1 ,y 2 ),w 2 (y 1 ,y 2 )] (y 1 ,y 2 )∈T

0elsewhere,

wherex 1 =w 1 (y 1 ,y 2 ),x 2 =w 2 (y 1 ,y 2 ) is the single-valued inverse ofy 1 =u 1 (x 1 ,x 2 ),
y 2 =u 2 (x 1 ,x 2 ). From this joint pmfpY 1 ,Y 2 (y 1 ,y 2 ) we may obtain the marginal pmf
ofY 1 by summing ony 2 or the marginal pmf ofY 2 by summing ony 1.
In using this change of variable technique, it should be emphasized that we need
two “new” variables to replace the two “old” variables. An example helps explain
this technique.

Example 2.2.1. In a large metropolitan area during flu season, suppose that two
strains of flu, A and B, are occurring. For a given week, letX 1 andX 2 be the
respective number of reported cases of strains A and B with the joint pmf

pX 1 ,X 2 (x 1 ,x 2 )=

μx 11 μx 22 e−μ^1 e−μ^2

x 1 !x 2!

,x 1 =0, 1 , 2 , 3 ,..., x 2 =0, 1 , 2 , 3 ,...,

and is zero elsewhere, where the parametersμ 1 andμ 2 are positive real numbers.

Thus the spaceSis the set of points (x 1 ,x 2 ), where each ofx 1 andx 2 is a non-

negative integer. Further, repeatedly using the Maclaurin series for the exponential

function,^3 we have

E(X 1 )=e−μ^1

∑∞

x 1 =0

x 1

μx 11

x 1!

e−μ^2

∑∞

x 2 =0

μx 22

x 2!

= e−μ^1

∑∞

x 1 =1

x 1 μ 1

μx 11 −^1

(x 1 −1)!

·1=μ 1.

Thusμ 1 is the mean number of cases of Strain A flu reported during a week.
Likewise,μ 2 is the mean number of cases of Strain B flu reported during a week.

(^3) See for example the discussion on Taylor series inMathematical Commentsas referenced in
the Preface.