Fundamentals of Probability and Statistics for Engineers

H ence, using Equation (5.25),

or

This is the so-called^2 distribution, to be discussed in more detail in Section
7.4.2.

Example 5.8.Problem: a random voltage 1 having a uniform distribution
over interval 90V V 1 110 V is put into a nonlinear device (a limiter), as
shown in Figure 5.14. Determine the probability distribution of the output
voltage V 2.
Answer: the relationship between 1 and 2 is, as seen from F igure 5.14,

95 105

2 (volts)

1 (volts)

1

2 = g(^1 )

Figure 5.14 Transformation defined by Equation (5.29)

Functions of Random Variables 133

 

ν

fY…y†ˆ

X^2

jˆ 1

fX‰g^ j^1 …y†Š

dg^ j^1 …y†

dy

(^)
(^)

ˆ

fX…

y^1 =^2 †

2 y^1 =^2

‡

fX…y^1 =^2 †

2 y^1 =^2

ˆ

1

… 2 y†^1 =^2

e^ y=^2 ;

fY…y†ˆ

1

… 2 y†^1 =^2

e^ y=^2 ; fory 0 ;

0 ; elsewhere:

8

<

:

… 5 : 28 †

 

V 2 ˆg…V 1 †; … 5 : 29 †

ν

ν

ν

V V

V