Page 166 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 149

where the integrals represent n-fold integrals with respect to the components of
x and y, respectively. Following the usual rule of change of variables in multiple
integrals, we can write (for example, see Courant, 1937):
Z Z Z Z

1
f x†dx ˆ f ‰g y†ŠjJjdy; 5:64†
X
X
R n R n
X Y
where J is the Jacobian of the transformation, defined as the determinant
qg qg qg

1
1
1
1 1 1

qy 1 qy 2 qy n

. .
. .
J ˆ . . : 5:65†

qg qg qg

1
1
1

n n n

qy 1 qy 2 qy n
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:

1
f y†ˆ f ‰g y†ŠjJj: 5:66†
X
Y
This result is stated as Theorem 5.4.
Theorem 5.4. For the transformation given by Equation (5.61) where X is a
continuous random vector and g is continuous with continuous partial deriva-
tives and defines a one-to-one mapping, the jpdf of Y, f (y), is given by
Y

1
f y† ˆ f ‰g y†ŠjJj; 5:67†
Y X
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 ˆ 1. Similarly, an extension is also
possible of Equation (5.24) for the n ˆ 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 transformation y ˆ g(x)admitsat

1

1
most a countable number of roots x 1 ˆ g (y), x 2 ˆ g (y), .... Then
1 2
r
X
1
f y†ˆ f ‰g y†ŠjJ j j; 5:68†
Y
X
j
jˆ1

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