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UNIVERSITY OF DUBLIN MA22S6 TRINITY COLLEGE Faculty of Science school of mathematics SF Science Trinity term, 2010 Course 22S6 Wednesday 12th May, 2010 Lower Luce Hall 9:30—11:30am Dr. Michael Peardon Attempt THREE questions. All questions carry 20 marks. Marks for each part of the question are shown in square brackets. Page 2 of 3 MA22S6 1. The theorem of conditional probability states that for two events A and B, P (A|B) = P (A ∩ B) P (B) (a) Explain what the expressions P (A|B) and P (A ∩ B) denote and use this result to show Bayes’ theorem: P (A) P (B|A) P (B) P (A|B) = [8] (b) A new experiment to determine the quality of a soil sample is devised. The experiment either rates the soil as “poor” or “good”. A “poor” sample is detected with probability 95% but the test has a 1% chance of mis-reporting a “good” sample as “poor”. On average, one soil sample per 100 is poor quality. The experiment returns a sample as “poor”; what is the probability this is in fact the case? [12] 2. {X1 , X2 , X3 , . . . Xn } is a sequence of n independent and identically distributed random numbers each with mean µ and variance σ 2 . (a) What does it mean to say the entries in the sequence are independent? Write an expression for E[Xi Xj ] when i 6= j. [2] (b) The sample mean is defined as n X̄ = 1X Xi . n i=1 2 Show σX̄ , the variance of X̄ is given by 2 σX̄ = 1 2 σ n [7] (c) What are the allowed possible values of X̄ and their probabilities when n = 3 and Xi takes values 2 with probability 31 Xi = −1 with probability 2 3 [7] (d) Describe the probability density function of X̄ as n gets very large, making sure to describe what happens to the expected value and mean of X. [4] Page 3 of 3 3. (a) Define the median MX of a sequence of n random samples, {X1 , X2 , X3 , . . . Xn }. MA22S6 [3] (b) If all elements of the sequence are independent and have the density function x + 1 −3 ≤ x ≤ 0 6 2 fX (x) = 1−x 0<x≤1 2 2 compute the expected value of X and the expected value of the median for a large sample. [12] (c) Describe the quicksort algorithm for sorting n numbers and explain how a modification of this method can be used to find the median efficiently. 4. (a) Describe what is meant by a Markov process. [5] [6] (b) A simple model describes the appearance of a genetic mutation from one generation to the next as a system that is in one of two possible states at random. χ1 corresponds to an individual with the mutation, χ2 corresponds to an individual without the mutation. The system undergoes transitions according to a Markov process with Markov matrix 0.9 0.3 M = 0.1 0.7 Compute the probabilty that an individual will have the mutation given its grand-parent has the mutation. [4] (c) Find the probability a member of the population has the mutation on average. c UNIVERSITY OF DUBLIN 2010 [10]