Fundamentals of Probability and Statistics for Engineers

5.6 The fo llowing 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

If X has the distribution

(a) Show that the PDF of Y, FY (y), is

(b) What is fY (y)?

5.7 The risk R of an accident for a vehicle traveling at a ‘constant’ speed V is given by

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)

are such that c (v 1 v 2 )/2.

5.8 Let Y g(X), with X uniformly distributed over the interval a x b. Suppose
that the inverse function X g 1 (Y ) is a single-valued function of Y in the interval
g(a) y g(b). Show that the pdf of Y is

where g (x) 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 F igure 5.23. Assuming that is uniformly distributed
from 0 to /2, determine the pdf of the projected area perpendicular to the stream.

Flow

Projected area

Plate with area a

Figure 5.23 Plate in flow stream, for Problem 5.9

Functions of Random Variables 155

YˆceX:

fX…x†ˆ

e^ x; forx 0 ;

0 ; elsewhere:



FY…y†ˆ^1

y

c

 

; foryc;

0 ; fory<c:

8

<

:

Rˆaeb…V^ c†

2

;

ˆ ‡

ˆ  

ˆ

 

fY…y†ˆ

1

b a

1

g^0 ‰g^1 …y†Š



; forg…a†yg…b†;

0 ; elsewhere:

8

<

:

(^0) ˆ



Θ
(v 1 ,v 2 )c,and )v 1 and (bv 2