Fundamentals of Probability and Statistics for Engineers

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 X Y < 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

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

is 1

3.25 Let X 1 ,X 2 ,...,Xn be independent and identically distributed random variables,
each with PDF FX (x). Show that

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 beordered from 1
(professional) to 7 (unskilled). Let ra ndom variable Xk denote the class order of the
kth generation. Then, for a given region, the following in formation is given:
(i) The pmf of X 0 is described by

(ii) The conditional probabilities and for

every k are given in Table 3.2.

Table 3. 2 for Problem 3.26

ij

1234567

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

Random Variables and Probability D istributions 73

 \

fX…x†ˆ

1

… 2 †^1 =^2

ex

(^2) = 2  2
; 1<x< 1 ;
fY…y†ˆ
1
… 2 †^1 =^2
ey
(^2) = 2  2
; 1<y< 1 :
ea^2 /2^2.
P‰min…X 1 ;X 2 ;...;Xn†uŠˆ 1 ‰ 1 FX…u†Šn;
P‰max…X 1 ;X 2 ;...;Xn†uŠˆ‰FX…u†Šn:
pX 0 91)ˆ 0 :00,pX 0 92)ˆ 0 :00, pX 0 93)ˆ 0 :04,
pX 0 94)ˆ 0 :06,pX 0 95)ˆ 0 :11,pX 0 96)ˆ 0 :28, andpX 0 97)ˆ 0 :51.
P 9 Xk‡ 1 ˆijXkˆj) fori,jˆ1, 2,..., 7
P 9 Xk‡ 1 ˆijXkˆj)