STAT 3004: Probability Models and Stochastic Processes 2021
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STAT 3004: Probability Models and Stochastic Processes
Final Exam, Semester 1, 2021
1. [8 Marks] Let (sn . n = 0. 1. 2. 7 7 7 ) be a branching process with s0 = 1 and with offspring distribution X ~ Bin(2. p) for some p ∈ (0. 1); that is,
贮(X = z) = ╱ ←z(2)px (1 - p)2 −x . z = 0. 1. 2 7
(a) Determine the probability generating function (pgf) of X, namely r(2) = 匝2X for 2 ∈ [0. 1].
(b) Determine explicit expressions for the mean and variance of sn
as a function of p, and determine through p when the process is sub-critical, critical, and super-critical.
(c) Determine the probability of ultimate extinction, denoted by 1 , as a function of p.
2. [8 Marks] Let X = (Xn . n = 0. 1. 7 7 7 ) be a Markov chain with state- space E = n1. 2. 3}, initial distribution 一(0) = (1. 0. 0), and one-step
transition matrix P = │ 0(0) . |
1 0 1 4 |
1(0)、 . 7 |
(a) Draw the transition diagram for this Markov chain.
(b) Find the three distinct eigenvalues 71 . 72 . 73 of P.
(c) Determine matrices R1 . R2 . R3 so that Pn = i(3)=1 7i(n)Ri , for
n = 0. 1. 7 7 7 .
(d) Calculate the probability that X3 = 1.
(e) Find the unique stationary (and limiting) distribution of the chain.
3. [8 Marks] An ant is in search of food, which appears on the floor ac- cording to a homogeneous spatial Poisson process with rate of 2 points per square meter.
(a) What is the probability that the ant finds n items of food within
a radius of r meters?
(b) What is the expected number and variance of the number of items
of food found by the ant within a radius of r meters?
(c) For 0 π s π r , m ∈ n0. 1. 7 7 7 . n}, and n ∈ n0. 1. 2. 7 7 7 }, determine the probability of m items of food being within radius s, given n items of food are within radius r. Identify this as a known distribution.
4. [8 Marks] Consider n machines maintained by a single machine repair robot. Each machine has an exponentially distributed lifetime with mean 2 weeks (i.e., before it fails). The machine repair robot begins work immediately when a machine fails, and works in the order that the machines have failed in the case of multiple machines failing. The ma- chine repair robot takes an exponentially distributed amount of time, with mean 2 days, to repair any machine. Initially, all the machines are working.
(a) Formulate a continuous-time Markov chain model for the problem
which counts the number of failed machines, specifying the state space E, the initial distribution 一(0) , and the Q-matrix Q.
(b) Draw the corresponding transition rate diagram.
(c) What is the long-run probability that no machines are under re- pair?
(d) Suppose one machine has failed and is under repair. What is the probability that the machine repair robot fixes the machine before another machine fails?
5. [8 Marks] Consider an abstract probability space (Ω . e. 贮). Answer the following questions.
(a) Let A and B be two disjoint subsets of Ω. Write down the smallest
A-algebra containing A and B .
(b) Let X1 . X2 . 7 7 7 be independent random variables on (Ω . e. 贮) with Xn ~ Ber(1/n). Does Xn a-s-.. 0 as n - o? Prove or disprove this.
(c) Let e1 . e2 . 7 7 7 . en be a collection of A-algebras on Ω. Show that the collection r defined by r = gk(n)=1ek is also a A-algebra of subsets of Ω.
2022-06-06