Question

U JOM U OL- 5.71 Two telephone calls come into a switchboard at random times in a fixed one-hour period. Assume that the calls are made independently of one another. What is the probability that the calls are made a b in the first half hour? within five minutes of each other?

Answer 1

5.71

From the information, observe that two telephone calls come into a switchboard at random times in a fixed one hour period. Assume that the calls are made independently of one another. The times are uniformly distributed over the one hour period (0,1) The marginal density functions are as follows: f(y)=1 12(y2)=1 If Y, and y, are independent, then f(x, y2) = f(y)/2 (y2) = 1x1 =1

a)

Calculate the probability that the calls are made in the first half hour. 0 0 PLY 50.5,, 50.5)= [] /<»»)v.dk - Ido do = | dv. (3) = 0.5| dv = 0.5(yz). =0.5(0.5) = 0.25

b)

Calculate the probability that the calls made within five minutes of each other. 1/12 -11/12 11/12 p(x-tisk, sy tina)= "7 lov dosi + ] tots dovi ="] v: () *"* + | dv. (2) +"| dv.() +1/12->)+ | «» (-+1/12) 11/12 =i[r]!" + izly: Ive 144' 144 22 144 = 0.1528