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Unformatted text preview: with respect to c yields
fY (c) = fX c−b
a 1
,
a (3.5) where we use the fact that a = −a for a < 0. Actually, (3.5) is also true if a > 0, because in that
case it is the same as (3.3). So (3.5) gives the pdf of Y = aX + b for any a = 0. Example 3.8.5 Suppose a vehicle is traveling in a straight line at constant speed a, and that a
random direction is selected, subtending an angle Θ from the direction of travel. Suppose Θ is
uniformly distributed over the interval [0, π ]. See Figure 3.15. Then the eﬀective speed of the
vehicle in the random direction is B = a cos(Θ). Find the pdf of B . 3.8. FUNCTIONS OF A RANDOM VARIABLE 103 B
Θ
a
Figure 3.15: Direction of travel and a random direction.
Solution: The range of a cos(θ) as θ ranges over [0, π ] is the interval [−a, a]. Therefore, FB (c) = 0
for c ≤ −a and FB (c) = 1 for c ≥ a. Let now −a < c < a. Then, because cos is monotone
nonincreasing on the interval [0, π ],
c
FB (c) = P {a cos(Θ) ≤ c} = P cos(Θ) ≤
a
−1 c
= P Θ ≥ cos
a
c
cos−1 ( a )
.
= 1−
π
1 Therefore,...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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