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Question 2 (20 marks) A hot spring resort arrange 5 sessions of hot spring experience tour every day. In each session, there could be a maximum of 15 visitors. Each visitor can visit different hot springs in this tour. Suppose the following table represents the probability distribution function of the number of hot springs (X) a visitor visits in a tour. X 1 2 3 4 P(X = x) 0.4 0.45 k k 2 (a) Find the value of k. (b) What is the probability that a visitor visits at least 2 hot springs in a tour? (c) Find E(X) and Var(X). (d) Suppose for every visit to a hot spring, the visitor will be offered two drinks. Use Y to denote the number of drinks a visitor receives in a tour. Calculate E(Y) and o(Y). (e) In this session, there are 15 visitors. Assume the number of visits made by visitors are independent. What is the probability that there are exactly 5 visitors visit at least 2 hot springs? (f) For many sessions, each with 15 visitors, find the expectation and standard deviation of the number of visitors visit at least 2 hot springs.