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Past Exam 1

MTH2222

Throughout this paper, CDF, PMF, PDF and MGF stand for cumulative distribution function, probability mass function, probability density function and moment generating function, respectively. All answers must be justified.

1 A laboratory blood test is effective in detecting a certain disease when it is in fact present, with probability 1 −α. However, the test also yields a “false positive” (i.e. the test result of a healthy person indicates he has the disease) with probability β. If in a given population, a randomly selected person actually has the disease with probability p, what is the probability a person has the disease given that his test result is positive?                       [4 marks]

2 Let X be a discrete random variable with PMF

where 0 < a < 1.

(a) Write down an expression for

(b) Obtain c, identify the distribution of X and deduce E[X] and var(X).                    [7 marks]

3 In a game show, contestants select at random (with equal probability) one of three topics, Anatomy, Biology and Chemistry. The host then asks n questions from the chosen topic. Marie’s knowledge allows her to correctly answer a randomly selected question from these topics with probability qA, qB and qC respectively. Assuming that, for a given topic, the outcomes of the n questions (correct or incorrect) are independent, find the PMF of the number of Marie’s correct answers, and its mean.                            [6 marks]

4 Let, for an integer k ≥ 3, X have the following PDF:

(a) Find c, E[X] and var(X).

(b) Find the PDF of ln X and identify its distribution.                                  [9 marks]

5 Let X be a random variable with MGF given by

Find the mean and variance of X.                                      [5 marks]

6 The joint PDF of X and Y is given by

where r > 0.

(a) Find the marginal PDF of X, and identify its distribution.

(b) Find the conditional PDF of Y given X = x.

(c) Deduce the conditional mean of Y given X = x.

(d) Find the mean and variance of Y .

(e) Find the covariance of X and Y by conditioning on X.

(f) Are X and Y independent? Explain.                                                           [17 marks]

7 Let X be a random variable with MGF given by

where n is a strictly positive integer.

(a) Identify the distribution of X when n = 1.

(b) Deduce the mean and variance of X for an arbitrary n.                              [6 marks]

8 Let X1, X2, . . . be a sequence of independent Bernoulli p random variables, let N be a binomial (n, a) random variable (i.e. with n trials and probability a of success) independent of the Xi ’s, and let

Obtain and identify the distribution of S.                                   [6 marks]

9 Let Xn be the number of successes in n Bernoulli trials where the probability of success is p. Let Yn = Xn/n be the average number of successes per trial. Derive the limit of Yn as n → ∞ by applying the Chebyshev inequality to the event {|Yn − p| > a}. In what sense does Yn converge to this limit?                                       [4 marks]