Page 165 - Fundamentals of Probability and Statistics for Engineers

148 Fundamentals of Probability and Statistics for Engineers

f (y)
Y

1
2
y
0 1 2 3

Figure 5.21 Probability density function, f (y), in Example 5.17

Y
The problem is to obtain the joint probability distribution of random variables
Y j , j ˆ 1, 2, .. . , m, which arise as functions of n jointly distributed random
variables X k , k ˆ 1, .. . , n. As before, we are primarily concerned with the case
in which X 1 ,..., X n are continuous random variables.
In order to develop pertinent formulae, the case of m ˆ n is first considered.
We will see that the results obtained for this case encompass situations in which
m < n.
Let X and Y be two n-dimensional random vectors with components
(X 1 ,..., X n ) and (Y 1 ,..., Y n ), respectively. A vector equation representing
Equation (5.60) is
Y ˆ g X† 5:61†

where vector g( X) has as components g 1 (X), g 2 ( X), ... g n (X). We first consider

the case in which functions g j in g are continuous with respect to each of their
arguments, have continuous partial derivatives, and define one-to-one

1
1
mappings. It then follows that inverse functions g of g , defined by

j

1
X ˆ g Y†; 5:62†
exist and are unique. They also have continuous partial derivatives.
In order to determine f ( y) in terms of f (x), we observe that, if a closed
X

Y
region R n in the range space of X is mapped into a closed region R n in the

X Y
range space of Y under transformation g, the conservation of probability gives
Z Z Z Z
f y†dy ˆ f x†dx; 5:63†
X
Y
R n Y R n X

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