Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

102 Multivariate Distributions

SinceFZ′(z) exists for all values ofz, the pmf ofZmay then be written

fZ(z)=

⎧

⎨

⎩

z 0 <z< 1

2 −z 1 ≤z< 2

0elsewhere.

(2.2.2)

In the last example, we used the cdf technique to find the distribution of the
transformed random vector. Recall in Chapter 1, Theorem 1.7.1 gave a transfor-
mation technique to directly determine the pdf of the transformed random variable
for one-to-one transformations. As discussed in Section 4.1 of the accompanying re-
sourceMathematical Comments,^4 this is based on the change-in-variable technique
for univariate integration. Further Section 4.2 of this resource shows that a simi-
lar change-in-variable technique exists for multiple integration. We now discuss in
general the transformation technique for the continuous case based on this theory.
Let (X 1 ,X 2 ) have a jointly continuous distribution with pdffX 1 ,X 2 (x 1 ,x 2 )and
support setS. Consider the transformed random vector (Y 1 ,Y 2 )=T(X 1 ,X 2 )where
Tis a one-to-one continuous transformation. LetT =T(S) denote the support of
(Y 1 ,Y 2 ). The transformation is depicted in Figure 2.2.1. Rewrite the transforma-
tion in terms of its components as (Y 1 ,Y 2 )=T(X 1 ,X 2 )=(u 1 (X 1 ,X 2 ),u 2 (X 1 ,X 2 )),
where the functionsy 1 =u 1 (x 1 ,x 2 )andy 2 =u 2 (x 1 ,x 2 ) defineT.Sincethetrans-
formation is one-to-one, the inverse transformationT−^1 exists. We write it as
x 1 =w 1 (y 1 ,y 2 ),x 2 =w 2 (y 1 ,y 2 ). Finally, we need theJacobianof the transfor-
mation which is the determinant of order 2 given by

J=

∣

∣

∣

∣

∣∣

∂x 1

∂y 1

∂x 1

∂y 2

∂x 2

∂y 1

∂x 2

∂y 2

∣

∣

∣

∣

∣∣.

Note thatJplays the role ofdx/dyin the univariate case. We assume that these
first-order partial derivatives are continuous and that the JacobianJis not identi-
cally equal to zero inT.
LetBbe any region^5 inT and letA =T−^1 (B) as shown in Figure 2.2.1.
Because the transformationTis one-to-one,P[(X 1 ,X 2 )∈A]=P[T(X 1 ,X 2 )∈
T(A)] =P[(Y 1 ,Y 2 )∈B]. Then based on the change-in-variable technique, cited
above, we have

P[(X 1 ,X 2 )∈A]=

∫∫

A

fX 1 ,X 2 (x 1 ,x 2 )dx 1 dy 2

=

∫∫

T(A)

fX 1 ,X 2 [T−^1 (y 1 ,y 2 )]|J|dy 1 dy 2

=

∫∫

B

fX 1 ,X 2 [w 1 (y 1 ,y 2 ),w 2 (y 1 ,y 2 )]|J|dy 1 dy 2.

(^4) See the reference forMathematical Commentsin the Preface.
(^5) Technically an event in the support of (Y 1 ,Y 2 ).