Page 150 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 133

Hence, using Equation (5.25),

2
1
X dg y†
j

1
f y†ˆ f ‰g y†Š
Y
X
j
jˆ1 dy
f
y 1=2 † f y 1=2 †
ˆ X ‡ X
2y 1=2 2y 1=2
1
y=2
ˆ e ;
2 y† 1=2
or
1
y=2
8
< e ; for y 0;
f y†ˆ 2 y† 1=2 5:28†
Y
:
0; elsewhere:
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 V 1 having a uniform distribution
over interval 90 V 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 V 1 and V 2 is, as seen from Figure 5.14,

V 2 ˆ g V 1 †; 5:29†

ν (volts)
2

ν = g( )
ν
2 1
1
ν (volts)
1
95 105

Figure 5.14 Transformation defined by Equation (5.29)

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