Question

Suppose that phone calls arrive at a switchboard according to a Poisson Process at a rate of 2 calls per minute.

(d)What is the probability that the next call comes in 30 seconds and the second call comes at least 45 seconds after that?

(e) Let T4 be the time between 1st and 5th calls. What is the distribution of T4?

(f) What is the probability that the time between 1st and 5th call is longer than 5 minutes?

Please shows steps.

Answer 1

Suppose that phone calls arrive at a switchboard according to a Poisson Process at a rate of 2 calls per minute.

d)

P(T1 < 30 sec , T2 > 45 sec) = P(T1 < 0.5 , T2 > 0.75)   

=P(T1 < 0.5) * P(T2 > 0.75)

Ti follow exponential distribution with rate = 2

P(T < t) = 1 - e^(- 2t)

P(T > t) = e^(- 2t)

P(T1 < 0.5) * P(T2 > 0.75) = (1 - e^(-0.5 *2)) * (e^(-0.75*2)

= 0.141045

e)

Let T4 be the time between 1st and 5th calls.

The distribution of T4 = n*p

= 5 * 4/5 = 4

f)

The probability that the time between 1st and 5th call is longer than 5 minutes is

P(T1 - T5 > 5) = [e^(-5 )*(5)^5]/5!

= 4.54/120

= 0.038

The probability that the time between 1st and 5th call is longer than 5 minutes is 0.038