Page 141 - Fundamentals of Probability and Statistics for Engineers

124 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

dF Y y† d
1
f y†ˆ ˆ f1
F X ‰g y†Šg
Y
dy dy
5:11†

1
dg y†

1
ˆ
f ‰g y†Š :
X
dy
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 .
1
Since the derivative dg (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 ˆ g(X ) where

g(X ) is continuous in X and strictly monotone. Then

1

1

f y†ˆ f ‰g y†Š dg y† ; 5:12†
X
Y
dy

where juj denotes the absolute value of u.

Example 5.3. Problem: the pdf of X is given by (Cauchy distribution):

a
f x†ˆ ;
1 < x < 1: 5:13†
X 2 2
x ‡ a †
Determine the pdf of Y where
Y ˆ 2X ‡ 1: 5:14†
Answer: the transformation given by Equation (5.14) is strictly monotone.
Equation (5.12) thus applies and we have
y
1

1
g y†ˆ ;
2
and

1
dg y† 1
ˆ :
dy 2

TLFeBOOK