Fundamentals of Probability and Statistics for Engineers

F ollowing Equation (5.12), the result is

It is valid over the entire range as it is in correspondence with the
range defined in the range space RX.

Example 5.4.Problem: the angle of a pendulum as measured from the
vertical is a random variable uniformly distributed over the interval
Determine the pdf of Y, the horizontal distance, as shown
in F igure 5.4.
Answer: the transformation equation in this case is

where

As shown in Figure 5.5, Equation (5.16) is monotone within the range

Hence, Equation (5.12) again applies and we have

Y

1

Figure 5.4 Pendulum, in Example 5.4

Functions of Random Variables 125

fY…y†ˆfX

y 

1

2



1

2



ˆ

a

2 

…y 

1 †^2

4 ‡a^2

"#

1

ˆ

2 a



1

…y 

1 †^2 ‡ 4 a^2

; 

1 <y< 1 :

… 5 : 15 †

1 <y< 1



1 <x< 1



 /2<</2).

Yˆtan; … 5 : 16 †

f…†ˆ

1



; for



2

<<



2

;

0 ; elsewhere.

8

<

:

… 5 : 17 †

/2<</2.

g^1 …y†ˆtan^1 y:

φ