Assignment 5 Material from Modules 7 & 8 Questions Total of marks is 38. Question 1 – 12 marks A supermarket has two customers waiting to pay for their purchases at counter I and one customer waiting to pay at counter II. Let Y1 and Y2 denote the numbers of customers who purchase more than $50 of groceries at the respective counters. Suppose that Y1 and Y2 are independent binomial random variables with the probability of a customer spending more than $50 equal to 0.2 for counter I and 0.3 for counter II. (a) Find the joint probability distribution for Y1 and Y2 [4 marks] (b) Find the probability that no more than one of the three customers spends in excess of $50. [4 marks] (c) Calculate the probability in (b) by simulation. [4 marks] (i) draw samples from the distributions of Y1 and Y2 (ii) count the frequencies of 0 and 1. Question 2 – 13 marks The random variable X has a gamma(; ) distribution; ie X has pdf fX(x) = x??1e??x=??()for x > 0: (a) Determine the pdf of Y = lnX and plot the density function. [5 marks] (b) Simulate 1000 values of Y and add the histogram of these to the above plot of the density function. [5 marks] (c) When is large, the gamma distribution is well approximated by the normal distribu- tion with the same mean and variance. Using this result, determine the approximate value of P(X