Question

There is a (μ=20)-Poisson number of customers X in a store at closing. Each customer will leave independently with rate 10/hr.

a) Show the number of customers leaving the store after closing is a Poisson measure N on (0,∞) and find its rate function.

b) What is the most likely number of customers 12 mins after closing?

Answer 1

Solution: Let X be the discrete random variable that represents the number of events observed over a given time period. Let be the expected value (average) of X. If X follows a Poisson distribution, then the probability of observing k events over the time period is

For example, the Poisson distribution is appropriate for modeling the number of phone calls an office would receive during the noon hour, if they know that they average 4 calls per hour during that time period.

Here in our case = 10

1. The Probability of N number of customers leaving the store after closing is given as ( N belongs to (0,∞))

To find number of customer leaving per hour , P(X=N)*(μ=20)