MTH2222 Past Exam 1
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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]
2023-06-10