Question

The number of incoming calls at a telephone exchange is modeled using a Poisson distribution with mean lambda= 2 per minute a What is the probability of having five or less calls in a 3 min interval b Show that given that there were in calls during the first t minutes the number of calls during the first S<t minutes follows a binomial with parameters n and s/t

Answer 1

a) As the mean number of calls is 2 per minute, therefore the mean number of calls in 3 minute interval would be given here as: 3*2 = 6

Therefore the probability here is computed as:
P(X <= 5)

This is computed in EXCEL here as:
=poisson(5,6,TRUE)

0.4457 is the output here.

Therefore 0.4457 is the required probability here.

b) Given that there are n calls in time t, the conditional distribution of the number of calls in s < t time is modelled here using Bayes theorem as:

Using the poisson probability function, we get here:

this is the the binomial distribution with parameters n and (s / t) here.