Question

On the average, every 12 seconds a customer makes a call using a certain phone card. Calls are modeled by a Binomial counting process with 2-second frames. Find the mean and the variance for the time, in seconds, between two consecutive calls.

Answer 1

Binomial counting process is applied and hence the inter arrival time follows geometric distribution since the inter arrival time is integer valued.

The geometric distribution with parameter p, where p is the probability of success has

Mean = E(X) =

1-р

Variance = V(X) =

1-р p2

Given that, on the average, every 12 seconds a customer makes a call using a certain phone card. Therefore probability of success, p = 112.

1. 12

Since it is given that the binomial process is in two second frames, the probability of success becomes p=16.

1 6

Therefore, the mean and variance of two consecutive calls are given by

Mean =

1 1. 6 1 = 5 6

Variance =

1 6 = 30 2 E)

The mean time, in seconds, between two consecutive calls is 5 seconds and the variance for between two consecutive calls is 30.