Page 148 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 131

where r is the number of roots for x of equation y ˆ g(x). Clearly, Equation
(5.12) is contained in this theorem as a special case (r ˆ 1).
Example 5.6. Problem: in Example 5.4, let random variable now be uni-
formly distributed over the interval
< < . Determine the pdf of
Y ˆ t an .
Answer: the pdf of is now
8
1
< ; for
< < ;
f †ˆ 2

0; elsewhere;
:
and the relevant portion of the transformation equation is plotted in
Figure 5.12. For each y, the two roots 1 and 2 of y ˆ tan are (see Figure
5.12)
9

1

1
1 ˆ g y†ˆ tan y; for
>
1 < 1 0 >
2 =
; y 0;
>

1

1
2 ˆ g y†ˆ tan y; for >
2 < 2 ;
2
9

1
1 ˆ tan y; >
2 =
for
< 1
>
; y > 0:
>

1
2 ˆ tan y; for 0 < 2 >
;
2
y
y

φ
π
π
– π φ – — φ — π
1 2 2 2

Figure 5.12 Transformation y ˆ tan

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