Two telephone calls come into a switchboard at random

Solution 71E

Step1 of 3:

Let us consider a random variable Y it presents the number of  telephone calls come into a switchboard. Assume that the calls are made independently of one another.

Here our goal is:

a). We need to find the probability that the calls are made in the first half hour.

b). We need to find the probability that the calls are made within five minutes of each other.

Step2 of 3:

a).

Here the times are uniformly distributed over the one hour period (0, 1):

Let us assume that  and  are independent, then:

           = 1

Therefore,

Then,