Question

A call centre can be modelled by a Poisson process with a rate of 1 call per minute.

1. What is the probability of no calls between 9 am and 9:05 am on a given morning?

2. Given that there have been no calls on a particular morning between 9 am and 9:05 am, what is the probability of exactly one call between 9:05 am and 9:10 am?

(Please attach a detailed explaination, thx)

Answer 1

Explanation:

Given 1 call per minute.

Poisson distribution formula is

1. probability of no calls between 9 am and 9:05 am on a given morning

9am to 9:05 am : 5 minutes.

So λ is 5.

No calls means x=0.

2.Given that there have been no calls on a particular morning between 9 am and 9:05 am, the probability of exactly one call between 9:05 am and 9:10 am. This can be calculated by using conditional probability.

P(a/b) = p(a and b)/p(b)

Let p(a) as 9:05 to 9:10 one call

Let p(b) as 9 to 9:05 no calls.

9:05 to 9:10 5 mins .

So lamdba is 5.

And having exactly one call so x=1.

This is p(a).

P(a/b) = p(a) since independent events.

The number of calls during 9 to 9:05 will not effect the calls during 9:05 to 9:10