Fundamentals of Probability and Statistics for Engineers

In the examples given above, it is easy to verify that all density functions
obtained satisfy the required properties.
Let us now turn our attention to a more general case where function
Y g(X) is not necessarily strictly monotonic. Two examples are given in
Figures 5.10 and 5.11. In Figure 5.10, the monotonic property of the transform-
ation holds for y < y 1 ,andy>y 2 , and Equation (5.12) can be used to
determine the pdf of Y in these intervals of y. For y 1 y , however, we
must start from the beginning and consider FY (y) P(Y y). The region
defined by Y y in the range space RY covers the heavier portions of the
function y g(x), as shown in Figure 5.10. Thus:

where are roots for x of function
()intermsof.
As before, the relationship between the pdfs of X and Y is found by differ-
entiating Equation (5.21) with respect to y. It is given by

x 1 =g 1 (y)

y=g(x)

x 2 =g 2 (y) x 3 =g 3 (y)

y 2

y

y

x

y 1

–1 –1 –1

Figure 5. 10 An example of nonmonotonic function y g(x)

Functions of Random Variables 129

ˆ

ˆ

FY…y†ˆP…Yy†ˆP‰Xg^11 …y†Š‡P‰g^21 …y†<Xg^31 …y†Š

ˆP‰Xg^11 …y†Š‡P‰Xg^31 …y†Š

P‰Xg^21 …y†Š

ˆFX‰g^11 …y†Š‡FX‰g^31 …y†Š

FX‰g^21 …y†Š; y 1 yy 2 ;

… 5 : 21 †

 y^2



ˆ 

ˆ

x 1 ˆg 1

1 y),x 2 ˆg^21 y) ,andx 3 ˆg^31 y)

yˆgx y

fY…y†ˆfX‰g^11 …y†Š
dg^11 …y†
dy
‡fX‰g^31 …y†Š
dg^31 …y†
dy
fX‰g^21 …y†Š
dg^21 …y†
dy
; y 1 yy 2 :

… 5 : 22 †