Fundamentals of Probability and Statistics for Engineers

Since derivative dg 21 (y) dy is negative whereas the others are positive,
Equation (5.22) takes the convenient form

Figure 5.11 represents the transformation y sin x; this equation has an infinite
(but countable) number of roots, x 1 g 11 (y), x 2 g 21 (y),... , for any y in the
interval 1 y 1. Following the procedure outlined above, an equation similar
to Equation (5.21) (but with an infinite number of terms) can be established for
FY (y) and, as seen from Equation (5.23), the pdf of Y now has the form

It is clear from Figure 5.11 that fY (y) 0 elsewhere.
A general pattern now emerges when function Y g(X) is nonmonotonic.
Equations (5.23) and (5.24) lead to Theorem 5.2.

Theorem 5. 2: Let X be a continuous random variable and Y g(X), where
g(X) is continuous in X, and y g(x) admits at most a countable number of
roots x 1 g 11 (y), x 2 g 21 (y),.... Then:

y

x

y

1

–1

Figure 5. 11 An example of nonmonotonic function y g(x)

130 Fundamentals of Probability and Statistics for Engineers

ˆ

(^) )/
fY…y†ˆ

X^3

jˆ 1

fX‰g^ j^1 …y†Š

dg^ j^1 …y†

dy

;y 1 yy 2 :… 5 : 23 †

fY…y†ˆ

X^1

jˆ 1

fX‰g^ j^1 …y†Š

dg^ j^1 …y†

dy

(^)
(^)
;
1 y 1 :… 5 : 24 †

ˆ

ˆ^ ˆ^

(^)  

ˆ

ˆ

ˆ

ˆ

ˆ^ ˆ^

fY…y†ˆ

Xr

jˆ 1

fX‰g^ j^1 …y†Š

dg^ j^1 …y†

dy

; … 5 : 25 †