Page 172 - Fundamentals of Probability and Statistics for Engineers

Functions of Random Variables 155

5.6 The following arises in the study of earthquake-resistant design. If X is the magni-
tude of an earthquake and Y is ground-motion intensity at distance c from the
earthquake, X and Y may be related by
X
Y ˆ ce :
If X has the distribution

x
e ; for x 0;
f x†ˆ
X
0; elsewhere:
(a) Show that the PDF of Y , F Y (y), is
8
y

1
; for y c;
<
F Y y†ˆ c
0; for y < c:
:
(b) What is f (y)?
Y
5.7 The risk R of an accident for a vehicle traveling at a ‘constant’ speed V is given by
R ˆ ae b V
c† 2 ;
where a, b, and c are positive constants. Suppose that speed V of a class of vehicles is
random and is uniformly distributed between v 1 and v 2 . Determine the pdf of R if (a)
(v , v ) c , and (b) v and v 2 are such that c ˆ (v 1 ‡ v 2 )/2.
1 2
1
5.8 Let Y ˆ g(X ), with X uniformly distributed over the interval a x b. Suppose

1
that the inverse function X ˆ g (Y ) is a single-valued function of Y in the interval
g(a) y g(b). Show that the pdf of Y is
1 1
8
< ; for g a† y g b†;
0

1
f y†ˆ b
a g ‰g y†Š
Y
:
0; elsewhere:
0
where g (x) ˆ dg(x)/dx.
5.9 A rectangular plate of area a is situated in a flow stream at an angle measured from
the streamline, as shown in Figure 5.23. Assuming that is uniformly distributed

from 0 to /2, determine the pdf of the projected area perpendicular to the stream.
Projected area
Flow
Plate with area a

Θ

Figure 5.23 Plate in flow stream, for Problem 5.9

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