Question 1 (10 marks)

Given the following discrete probability distribution:

X 1 2 4 5 9 15

P(X) 0.13 0.18 0.23 0.31 0.09 0.06

(a)Compute the expected value and the variance for this distribution.

(b)Determine the median and the mode for this distribution. Comment on the shape of this distribution with a reason to support your conclusion. (4 marks)

(c)Given that a randomly selected value of X is at most 5, what is the probability that it is an even number?

Question 2 (20 marks)

In a supermarket, there are two cashier counters: a regular counter and an express counter. It is known that 88% of customers pay at the regular counter. It is found that the waiting time for a customer to pay at the regular counter follows the normal distribution with mean 6.6 minutes and standard deviation 1.2 minutes.

(a)Find the probability that the waiting time for a customer to pay at the regular counter is more than 6 minutes. (3 marks)

(b)Suppose 12 customers who pay at the regular counter are randomly selected. Assume that their waiting times are independent. Find the probability that there are more than 10 of the 12 customers each having a waiting time of more than 6 minutes. (4 marks)

It is found that the waiting time for a customer to pay at the express counter follows the normal distribution with mean????minutes and standard deviation 0.8minutes. It is known that exactly 21.19% of the customers pay at the regular counter wait less than????minutes, while exactly 3.59% of the customers at the express counter wait more than????minutes.

(c) Find????and????. (4 marks)

(d) For a randomly selected customer, given that he waits more than????minutes to pay, find the probability that he pays at the regular counter. (9 marks)