Answer Happy

Soumya is a newly employed statistician working for a car insurance company. He is analysing data which records the number of cars registered to clients in the Cotswolds. Soumya summaries the company's data to create a table of probabilities, given below, for the random variable X which is the number of cars registered to each client. T 1 2 3 4 p(x) 0.25 0.54 0.19 0.12 (a) Show that Soumya made a mistake when extracting the probabilities from the data, by showing that the table does not satisfy the conditions for a valid p.m.f. Make sure you investigate all the requirements for a valid p.m.f. and report on each of these.
(b) When Soumya looked at the data again he realised that the probability he calculated for any client to register 3 cars was incorrect. Assuming that all the probabilities, except p(3) in the table above are correct, calculate the correct probability for p(3) and produce a correct version of the table showing the p.m.f.
(c) Using your answer to part (b), show that the mean of X is 2.08.
(d) Using your answer to part (b) calculate the variance of X, correct to two decimal places.