Fundamentals of Probability and Statistics for Engineers

where r is the number of roots for x of equation y g(x). Clearly, Equation
(5.12) is co ntained in this theorem as a sp ecial case (r 1).

Example 5.6.Problem: in Example 5.4, let random variable now be uni-
formly distributed over the interval. Determine the pdf of
Ytan.
Answer: the pdf of is now

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 tan are (see Figure
5.12)

y

y

Figure 5. 12 Transformation y tan

Functions of Random Variables 131

ˆ

ˆ

<<





f…†ˆ

1

2 

; for <<;

0 ; elsewhere;

8

<

:

  

 1 ˆg^11 …y†ˆtan^1 y; for



2

< 1  0

 2 ˆg^21 …y†ˆtan^1 y; for



2

< 2 

9

>>

=

>>

;

; y 0 ;

 1 ˆtan^1 y; for < 1 



2

 2 ˆtan^1 y; for 0 < 2 



2

9

>>

=

>>

;

; y> 0 :

ππ

φ



ˆ

ˆ

φ 1 φ 2 —π

2

—π

2

––

ˆ