Question 4: (40 marks; 40 minutes) Answer each of the following. (a) A single multiple choice question on a test has" choices, only one of which is correct. A student who does not know the answer will guess randomly, while a student who does know the answer will answer correctly. The probability that a student taking this test knows the answer is?. Find the conditional probability that the student knew the answer, given that the question was answered correctly. (b) Let X > have a Poisson distribution with parameter > > . Suppose that A itself is random, following an exponential distribution with parameter. What is the conditional distribution of A|X = x? (c) Let continuous random variable ^ have a uniform distribution over the range [-1, 1] and let Y = X3. Prove that X and Y are uncorrelated. Does this imply that they are independent random variables? (d) Explain and discuss the Bayesian method of estimation of the parameters of a statistical model