Question

Customers arrive at a service facility according to a Poisson process of rate 5/hour. Let N(t) be the number of customers that have arrived up to time t (t hours) a. What is the probability that there is at least 2 customer walked in 30 mins? b. If there was no customer in the first 30 minutes, what is the probability that you have to wait in total of more than 1 hours for the 1st customer to show up? For any random customer, if there is 50% chance the customer is female, what is the expected waiting time until the Sth female customer comes in? C.

Answer 1

a)

for 30 min, $\lambda$ = 5/2 = 2.5

P(X>= 2) = 1- P(X =0) - P(X = 1)

= 1 -e^(-2.5) (1 + 2.5)

=0.71270250481

b)

P( X = 0 in 1 hr | X = 0 in 30 min)

= P( X = 0 in 1 hr and X = 0 in 30 min) /P(X = 0 in 30 min)

= P( X = 0 in next 30 min and X = 0 in 30 min) / P( X = 0 in 30 min)

= P( X = 0 in next 30 min )

= e^(-2.5)