Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 73Assume that X and Y are independent. Determine P(X 25 \ Y > 8), the probabilitythat the next earthquake within 50 miles will have a magnitude greater than8 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 thesquare (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 guidanceinaccuracies, atmospheric disturbances, and other error sources. Taking the originof the coordinates as the designed point of impact, the X and Y coordinates of theactual impact point are random, with marginal density functions1f X …x† ˆ…2† e x2 =2 2 ; 1 < x < 1;1=21f Y …y† ˆ…2† e y2 =2 2 ; 1 < y < 1:1=2Assume that the random variables are independent. Show that the probabilityof a splashdown lying within a circle of radius a centered at the originis 1 e a2 /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 thatP‰min…X 1 ; X 2 ; ...; X n †uŠ ˆ1 ‰1 F X …u†Š n ;P‰max…X 1 ; X 2 ; ...; X n †uŠ ˆ‰F X …u†Š n :The above are examples of extreme-value distributions. They are of considerablepractical 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 thekth generation. Then, for a given region, the following information is given:(i) The pmf of X 0 is described by p X01) ˆ 0:00, p X02) ˆ 0:00, p X03) ˆ 0:04,p X04) ˆ 0:06, p X05) ˆ 0:11, p X06) ˆ 0:28, and p X07) ˆ 0:51.(ii) The conditional probabilities PX k‡1 ˆ ijX k ˆ j) for i, j ˆ 1, 2, ..., 7 and forevery k are given in Table 3.2.Table 3. 2 PX k‡1 ˆ ijX k ˆ j) for Problem 3.26ij1 2 3 4 5 6 71 0.388 0.107 0.035 0.021 0.009 0.000 0.0002 0.146 0.267 0.101 0.039 0.024 0.013 0.0083 0.202 0.227 0.188 0.112 0.075 0.041 0.0364 0.062 0.120 0.191 0.212 0.123 0.088 0.0835 0.140 0.206 0.357 0.430 0.473 0.391 0.3646 0.047 0.053 0.067 0.124 0.171 0.312 0.2357 0.015 0.020 0.061 0.062 0.125 0.155 0.274TLFeBOOK