Mathematical Probability (STAT2003/STAT7003) Problem Set 3 The due date/time is given on Blackboard. STAT7003 students have an additional ques- tions [2(f), 3(e) and 4(c)] marked with a star (*). 1. Let X1 and X2 be two independent random variables with the probability density function of Xi given by f(x) = 1 x ln 2 , x 2 (0.5, 1) 0, else. (a) Determine the probability density function of Y = X1X2. [6 marks] (b) Determine the probability density function of Z = Y, Y 2 (0.5, 1) 2Y, Y 2 (0.25, 0.5] [3 marks] 2. (a) Let U and V be two independent random variables such that U Ber(1 %) with % 2 [0, 1) and V Exp() with > 0. Determine the moment generating function of W = UV . [3 marks] (b) Let {Wi}1i=1 be a sequence of independent and identically distributed random variables where Wi has the same distribution as W from part (a). Let X0 be a random variable independent of the {Wi}1i=1 and define the sequence of random variables {Xn}1n=0 recursively as Xn = %Xn1 +Wn. Show that if X0 Exp(), then Xn Exp() for all n > 0. [3 marks] (c) Determine Cov(Xn+k, Xn) for all n and k > 0. [2 marks] (d) Determine E(Xn |Xn1 = x) and Var(Xn |Xn1 = x). [2 marks] (e) Simulate a sample path for n = 0, . . . , 50 with % = 0.75 and = 1. [2 marks] (f) * Let {Yn}1n=0 be the sequence of random variables such that Y0 Gamma(2,) for some > 0, and Yn = %Yn1 +fWn, where {fWn}1n=1 is a sequence of independent and identically distributed ran- dom variables and Y0 is independent of the {fWn}1n=1. Determine the moment generating function of the {fWn}1n=1 such that Yn Gamma(2,) for all n > 0. Describe how fWn can be simulated using random variables with Bernoulli and Exponential distributions. [3 marks] 1 3. Consider the following system comprised of three components: The system is working if there is a path from left to right through working compo- nents. Components fail independently and the time to failure for each component has an exponential distribution with a mean of one year. (a) Determine an expression for the probability that the system is working at time t. [3 marks] (b) Determine the mean time to failure for the system. [2 marks] (c) Determine the probability that component two in the system is still working at time t given the system is working at time t. What is the limiting value as t!1 [3 marks] (d) Determine the failure rate for the system. [1 mark] (e) * Show the system has an increasing failure rate. [2 marks] 4. Consider the 4-state Markov chain X = {X1, X2, . . .} described by the following transition graph. 1 2 3 412 1 2 1 1 1 (a) Determine ( , , ) such that the limiting distribution of theX is = ( 110 , 2 10 , 3 10 , 4 10). [4 marks] 2 (b) Let ( , , ) = (13 , 1 4 , 1 5). Using a grid like the one below, sketch by hand a typical realisation of Xn, n = 1, . . . , 30, where X1 = 1. [2 marks] 5 10 15 20 25 30 1 2 3 4 (c) * For ( , , ) = (12 , 1 2 , 1 2) the limiting distribution of the Markov chain X is = (14 , 1 4 , 1 4 , 1 4). Define the sequence of random variables {Yn}10n=1 such that Yn = X11n. Show that Y = {Y1, Y2, . . . , Y10} is a Markov chain and determine the matrix of one-step transition probabilities for Y . [5 marks] 5. Let X1, . . . , Xn be independent random variables where the Xi have a Geo(p) distri- bution. Define Sn = X1 +X2 + · · ·+Xn. (a) Show that for any a > 1 and any t 2 [0, ln(1 p)), P(Sn > an/p) 6 pnen(1a/p)t (1 (1 p)et)n =: H(t; a). [2 marks] (b) As this upper bound H(t; a) holds for all t 2 [0, ln(1 p)), the tightest upper bound is found by minimising H(t; a) over t. For a fixed value of a, find the value ta which minimises H(t; a). [2 marks] 3
