suppose that customers arrive at a store according to a pois

suppose that customers arrive at a store according to a poisson process with rate one customer per minute. suppose that time t=0 is 8am t in minutes.

a) find the probability that exactly three customers arrive from 8 a.m to 8.05am.

b) find the probability of the single event that excatly 3 customers arrive between 8 a.m and 8.02 am and excatly 2 customes arrive between 8.01 am and 8.04 am.

c) find the probability that the fourth customer arreives by 8.05 am.

d) its known that excatly 6 customers have arrived by 8.10am. find the variance of the arrival time point of the fourth customer.

Solution

Average customer per minute = 1

a) Prob (3 customers in 5 minutes ) = 0.1404

b) P(3 customers in 2 minutes) x P(2 in 4 minutes)

=0.1804(0.1465)(since independent)

= 0.0264

c) P( 4th customer at 8.05)

= P(3 customers in 4 minutes)(p 1 customer at last 1 minute)

= 0.1954(0.3679)

= 0.0719

d) 6 customers arrive in 10 minutes

Gap between two cusomers = 1 minute

Hence fourth customer can arrive in 1, 2, 3,4,5,6, 7,8,9 or 10th minute

d)