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ST2351 Probability and Theoretical Statistics Tutorial 2: Discrete Random Variables 1. A fair coin (with sides “H” and “T”) is thrown twice. (a) Write down the sample space of the experiment. (b) The random variable X is defined to be: X(HH) = 2, X(HT ) = X(T H) = 1, X(T T ) = 0. Check that X satisfies the definition of a random variable. 2. You are sitting 6 exams. Your probability of passing each exam is 0.8, and each exam is passed independently. (a) What type of distribution describes the number of exams that you pass? (b) Calculate P (pass 4 or fewer exams). (c) Suppose that you retake a failed exam until you pass it, then continue on to take the next exam. i. What probability distribution describes the number of the times you must take an exam until you pass it? ii. What is the probability that you first pass an exam on the third attempt? iii. What is the distribution of the total number of attempts to pass all 6 exams? What is the probability that it takes 10 attempts? 3. The number of customers to enter a bank each minute is Poisson distributed with λ = 2. (a) Calculate P (1 customer enters). (b) What is P (3 or more enter)? 4. X is a random variable on {0, 1, 2, 3, 4} with probability mass function: p(0) = 0.1, p(1) = 0.3, p(2) = 0.05, p(3) = 0.25, p(4) = 0.3. Compute the expectation and variance of X. Also compute E(X 3 ). 5. X is a random variable with mass function pX (x) = 6 , x = 1, 2, . . . . x2 π2 Show that the mean value of X does not exist. 6. (Poisson approximation to the binomial). A telephone exchange in a city has 10000 subscribers. In each minute, there is a small probability, 0.0001, that a subscriber will make a call. Assume that each subscriber makes calls independently. (a) What is the probability distribution of the number of calls made to the exchange per minute? (b) What is the mean number of calls per minute? (c) Write down the probability distribution for the Poisson distribution that has the same mean. (d) Calculate P (0 calls), P (1 call), P (2 calls) and P (3 calls) for the binomial distribution in (a) and the Poisson in (c). Are the probabilities close? 1 c Simon Wilson, 2013