Explain briefly why it might be reasonable to expect these calls to arrive at random.

The call centre of a major energy provider expects to be particularly busy during the early shift in the early Winter. During the early shift calls arrive at a mean rate of 240 per hour and are believed to arrive at random. :

i) Explain briefly why it might be reasonable to expect these calls to arrive at random.

ii) What would be the probability distribution of the number of calls arriving during a five minute period in the early shift?( There is no need to calculate any probabilities in this part)

iii) Show(using probability tables or probability formulae) that the probability that there are more than 26 calls in a five-minute period is 0.0778. ***Show your working***

iv) Staffing levels during the early shift are such that they can cope with occasional peaks in arrivals, but service levels deteriorate rapidly when they experience a number of peaks close together. Continuing to assume that calls arrive at random, show that the probability that there are 6 or more five-minute periods in an hour in which the number of calls exceeds 26 is less than 1 in 1000. Show your working.

(v) Quiet periods only occur very rarely during the early shift in the call centre, so when gaps between calls exceed 2 minutes the management takes it as a signal of a telephone system failure and resets the system. What is the chance they reset the system unnecessarily in response to a 2 minute gap? Justify your method.

(vi) You have seen in class how a Normal distribution can be used to approximate a Binomial distribution under certain conditions on n and p. (Remember that the Normal distribution was chosen to match the Binomial distribution in mean and standard deviation A Normal distribution can also be used to approximate a Poisson distribution under certain conditions. By applying the same idea, use the Normal distribution to provide an approximate answer to part (iii) above. Explain your method carefully.

(vii) Suggest the conditions under which a Normal distribution can be used to approximate a Poisson distribution. Justify your answer.

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