2. The Poisson distribution with parameter A approximates the binomial distribution with parameters (n, p) when n ~ 00, p ~ 0, and np ~ A. Demonstrate this result for the situation where a manufactured lot is known to contain 1% defective items. If a sample of 10 items is taken from the lot, compute the probability of at most one defective item in a sample, first by using the (exact) binomial distribution and then by using the (approximate)

Poisson distribution. Show that the approximation will not be acceptable if the value of p is increased to, say, 0.5.

Customers arrive randomly at a checkout counter at the average rate of 20 per hour.

(a) Determine the probability that the counter is idle.

(b) What is the probability that at least two people are in line awaiting service?

Prove the formulas for the mean and variance of the Poisson distribution.