Fundamentals of Probability and Statistics for Engineers

The problem is to obtain the joint probability distribution of random variables
Yj, j 1,2,...,m, which arise as functions of n jointly distributed random
variables Xk, k 1,...,n. As before, we are primarily concerned with the case
in which X 1 ,...,Xn are continuous random variables.
In order to develop pertinent formulae, the case of m 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 ,...,Xn)and(Y 1 ,...,Yn), respectively. A vector equation representing
Equation (5.60) is

where vector g( X) has as components g 1 (X), g 2 ( X),...gn(X). We first consider
the case in which functions gj in g are continuous with respect to each of their
arguments, have continuous partial derivatives, and define one-to-one

mappin gs. It then follows that in verse functions g (^) j^1 of g^1 , defined by
exist and are unique. They also have continuous partial derivatives.
In order to determine fY ( y)intermsoffX (x), we observe that, if a closed
region RnXin the range space of X is mapped into a closed region RnY in the
range space ofYunder transformationg, the conservation of probability gives
0 123
1
2
fY(y)
y
Figure 5. 21 Probability density function, fY (y), in Example 5.17
148 Fundamentals of Probability and Statistics for Engineers

ˆ

ˆ

ˆ

Yˆg…X†… 5 : 61 †

Xˆg^1 …Y†;… 5 : 62 †

Z



Z

RnY

fY…y†dyˆ

Z



Z

RnX

fX…x†dx; … 5 : 63 †