(a) Give two conditions which must apply when modelling a random variable by a Poisson distribution. A particular make of kettle is sold by a shop at an average rate of five per week. The random variable X represents the number of kettles sold in any one week and X is modelled by a Poisson. distribution. The shop manager notices that at the beginning of a particular week there are seven kettles in stock.

(b) Find the probability that the shop will not be able to meet all the demands for kettles that week, assuming that it is not possible to. restock during the week. In order to increase sales performance, the manager decides to have in stock at the beginning of each week sufficient kettles to have at least a 99% chance of being able to meet all demands during that week.

(c) Find the smallest number of kettles that should be in stock at the beginning of each week.

(d) Using a suitable approximation find the probability that the shop sells at least 18 kettles in a four-week period, subject to stock always being available to meet demand. (L)