Page 90 - Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 73

Assume that X and Y are independent. Determine P(X 25 Y > 8), the prob-
\
ability that the next earthquake within 50 miles will have a magnitude greater than
8 and that its epicenter will lie within 25 miles of the nuclear plant.
3.23 Let random variables X and Y be independent and uniformly distributed in the
square (0, 0) < (X , Y ) < (1, 1). Determine the probability that XY < 1/2.
3.24 In splashdown maneuvers, spacecrafts often miss the target because of guidance
inaccuracies, atmospheric disturbances, and other error sources. Taking the origin
of the coordinates as the designed point of impact, the X and Y coordinates of the
actual impact point are random, with marginal density functions
1 2 2
f x†ˆ e x =2 ; 1 < x < 1;
X 1=2
2 †
1 2 2
f y†ˆ 1=2 e y =2 ; 1 < y < 1:
Y
2 †
Assume that the random variables are independent. Show that the probability
of a splashdown lying within a circle of radius a centered at the origin
2
is 1 e a /2 2 .
3.25 Let X 1 , X 2 , ..., X n be independent and identically distributed random variables,
each with PDF F X (x). Show that

n
P‰min X 1 ; X 2 ; ... ; X n † uŠˆ 1 ‰1 F X u†Š ;
n
P‰max X 1 ; X 2 ; ... ; X n † uŠˆ‰F X u†Š :
The above are examples of extreme-value distributions. They are of considerable
practical importance and will be discussed in Section 7.6.
3.26 In studies of social mobility, assume that social classes can be ordered from 1
(professional) to 7 (unskilled). Let random variable X k denote the class order of the
kth generation. Then, for a given region, the following information is given:
(i) The pmf of X 0 is described by p 91) ˆ 0:00, p 92) ˆ 0:00, p 93) ˆ 0:04,
X 0 X 0 X 0
p 94) ˆ 0:06, p 95) ˆ 0:11, p 96) ˆ 0:28, and p 97) ˆ 0:51.
X 0 X 0 X 0 X 0
(ii) The conditional probabilities P9X k‡1 ˆ ijX k ˆ j) for i, j ˆ 1, 2, ... , 7 and for
every k are given in Table 3.2.

Table 3.2 P9X k‡1 ˆ ijX k ˆ j) for Problem 3.26
i j
1 2 3 4 5 6 7

1 0.388 0.107 0.035 0.021 0.009 0.000 0.000
2 0.146 0.267 0.101 0.039 0.024 0.013 0.008
3 0.202 0.227 0.188 0.112 0.075 0.041 0.036
4 0.062 0.120 0.191 0.212 0.123 0.088 0.083
5 0.140 0.206 0.357 0.430 0.473 0.391 0.364
6 0.047 0.053 0.067 0.124 0.171 0.312 0.235
7 0.015 0.020 0.061 0.062 0.125 0.155 0.274

TLFeBOOK