In a one minute interval, the number of requests for access to a web-based game is a Poisson random variable with a mean of 20.

(a) Find the probability that more than 25 requests are received in the one minute interval by applying the central limit theorem. (Hint: apply the de Moivre-Laplace approximation)

(b) Calculate the exact probability in part (a) using the Poisson random variable. Compare with the result you obtained in part (a). (Hint: use Matlab to code the Poisson PMF)

(c) The web game server has a capacity for C requests per minute. If the number of requests exceed C, the server is overloaded. Apply the central limit theorem to estimate the smallest value of C for which the probability of overload is less than 0.05.

(d) Use MATLAB to calculate the actual probability of overload for the value of C derived from the central limit theorem in part (c).

(e) For the value of C derived from the central limit theorem, what is the probability of overload in a one-second interval?

(f) Comment on the application of the central limit theorem to estimate the overload probability in the one-second interval versus that in the one-minute interval.