Page 143 - Fundamentals of Probability and Statistics for Engineers

126 Fundamentals of Probability and Statistics for Engineers

y

y = tan φ

φ
π 0 π
− — —
2 2

Figure 5.5 Transformation defined by Equation (5.16)

and

1
dg y† 1
ˆ :
dy 1 ‡ y 2
The pdf of Y is thus given by

f tan
1 y†

f y†ˆ
Y 2
1 ‡ y
5:18†
1
ˆ ;
1 < y < 1:
1 ‡ y †
2
The range space R Y corresponding to
/2 < < /2 is
1 < y < 1. The
pdf given above is thus valid for the whole range of y. The random variable Y
has the so-called Cauchy distribution and is plotted in Figure 5.6.
Example 5.5. Problem: the resistance R in the circuit shown in Figure 5.7 is

random and has a triangular distribution, as shown in Figure 5.8. With a
0 1 A and a constant resistance r 0 ˆ

constant current i ˆ : 100 ; determine the
pdf of voltage V.
Answer: the relationship between V and R is
V ˆ i R ‡ r 0 †ˆ 0:1 R ‡ 100†; 5:19†

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