Fundamentals of Probability and Statistics for Engineers

and

The pdf of Y is thus given by

The range space RY corresponding to is 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.

Ex ample 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
constant current i 0 1A and a constant resistance r 0 100 ; determine the
pdf of voltage V.
Answer: the relationship between V and R is

Figure 5.5 Transformation defined by Equation (5.16)

126 Fundamentals of Probability and Statistics for Engineers

y

= tanφ

−— 2 π —π 2

0 φ

y

dg^1 …y†

dy

ˆ

1

1 ‡y^2

:

fY…y†ˆ

f…tan^1 y†

1 ‡y^2

ˆ

1

… 1 ‡y^2 †

; 

1 <y< 1 :

… 5 : 18 †

/2<</2 

1 <y< 1.

ˆ : ˆ

Vˆi…R‡r 0 †ˆ 0 : 1 …R‡ 100 †; … 5 : 19 †