Page 142 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 125

Following Equation (5.12), the result is
y
1 1

f y†ˆ f
Y X
2 2

1
" 2 #
a y
1†
ˆ 5:15†
2 4 ‡ a 2
2a 1
ˆ ;
1 < y < 1:
2
y
1† ‡ 4a 2
It is valid over the entire range
1 < y < 1 as it is in correspondence with the
range
1 < x < 1 defined in the range space R X .
Example 5.4. Problem: the angle of a pendulum as measured from the
vertical is a random variable uniformly distributed over the interval

/2 < < /2). Determine the pdf of Y , the horizontal distance, as shown
in Figure 5.4.
Answer: the transformation equation in this case is

Y ˆ tan ; 5:16†
where
1
8
; for
< < ;
<
f †ˆ 2 2 5:17†

0; elsewhere.
:
As shown in Figure 5.5, Equation (5.16) is monotone within the range

/2 < < /2. Hence, Equation (5.12) again applies and we have

1
g y†ˆ tan
1 y:

φ
1

Y
Figure 5.4 Pendulum, in Example 5.4

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