where the integrals represent n-fold integrals with respect to the components of
xandy, respectively. Following the usual rule of change of variables in multiple
integrals, we can write (for example, see Courant,1937):
where J is the Jacobian of the transformation, defined as the determinant
As a point of clarification, let us note that the vertical lines in Equation (5.65)
denote determinant and those in Equation (5.64) represent absolute value.
Equations (5.63) and (5.64) then lead to the desired formula:
This result is stated as Theorem 5.4.
Theorem 5.4.For the transformation given by Equation (5.61) whereXis a
continuous random vector andgis continuous with continuous partial deriva-
tives and defines a one-to-one mapping, the jpdf of Y,fY(y), is given by
where J is defined by Equation (5.65).
It is of interest to note that Equation (5.67) is an extension of Equation
(5.12), which is for the special case of n 1. Similarly, an extension is also
possible of Equation (5.24) for the n 1 case when the transformation admits
more than one root. Reasoning as we have done in deriving Equation (5.24), we
have Theorem 5.5.
Theorem 5.5.In Theorem 5.4, suppose transformationy g(x)admitsat
most a countable number of rootsx 1 g 11 (y),x 2 g 21 (y),.... Then
Functions of Random Variables 149
Z
Z
RnXfX
xdxZ
Z
RnYfXg^1
yjJjdy;
5 : 64 J
qg^11
qy 1qg^11
qy 2qg^11
qyn
... ...
qg^ n^1
qy 1qg^ n^1
qy 2qg^ n^1
qyn(^)
(^)
:
5 : 65
fY
yfXg^1
yjJj:
5 : 66
fY
yfXg^1
yjJj;
5 : 67
fY
yXrj 1fXg^ j^1
yjJjj;
5 : 68 (^)