Fundamentals of Probability and Statistics for Engineers

In comparison with Equation (5.7), Equation (5.10) yields a different relation-
ship between the PDFs of X and Y owing to a different g(X).
The relationship between the pdfs of X and Y for this case is again obtained
by differentiating both sides of Equation (5.10) with respect to y, giving

Again, we observe that Equations (5.10) and (5.11) hold for all continuous g(x)
that are strictly monotonic decreasing functions of x, that is g(x 2 )< g(x 1 )
whenever x 2 >x 1.
Since the derivative dg^1 (y)/dy in Equation (5.8) is always positive – as g(x) is
strictly monotonic increasing – and it is always negative in Equation (5.11) – as
g(x) is strictly monotonic decreasing – the results expressed by these two
equations can be combined to arrive at Theorem 5.1.

Theorem 5. 1. Let X be a continuous random variable and Y g(X) where
g(X) is continuous in X and strictly monotone. Then

where denotes the absolute value of u.

Ex ample 5. 3. Problem: the pdf of X is given by (Cauchy distribution):

Determine the pdf of Y where

Answer: the transformation given by Equation (5.14) is strictly monotone.
Equation (5.12) thus applies and we have

and

124 Fundamentals of Probability and Statistics for Engineers

fY…y†ˆ

dFY…y†

dy

ˆ

d

dy

f 1

 FX‰g^1 …y†Šg

ˆ

fX‰g^1 …y†Š

dg^1 …y†

dy

:

… 5 : 11 †

ˆ

fY…y†ˆfX‰g^1 …y†Š

dg^1 …y†

dy

; …^5 :^12 †

fX…x†ˆ

a

…x^2 ‡a^2 †

; 

1 <x< 1 : … 5 : 13 †

Yˆ 2 X‡ 1 : … 5 : 14 †

juj

g^1 …y†ˆ

y 

1

2

;

dg^1 …y†

dy

ˆ

1

2

: