STAT3004 Probability Models and Stochastic Processes Semester 1, 2021
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STAT3004
Probability Models and Stochastic Processes
Semester One Final Examinations, 2021
1. [8 Marks] Let (Sn , n = 0, 1, 2, . . . ) be a branching process with S0 = 1 and with offspring distribution X ~ Bin(2, p) for some p ∈ (0, 1); that is,
P(X = x) = ╱ ←x(2)px (1 - p)2 −x , x = 0, 1, 2 .
(a) Determine the probability generating function (pgf) of X, namely r(z) = 匝zX for z ∈ [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 η , as a function of p.
2. [8 Marks] Let X = (Xn , n = 0, 1, . . . ) 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)、 . . |
(a) Draw the transition diagram for this Markov chain.
(b) Find the three distinct eigenvalues λ 1 , λ2 , λ3 of P.
(c) Determine matrices R1 , R2 , R3 so that Pn = i(3)=1 λi(n)Ri , for
n = 0, 1, . . . .
(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, . . . , n}, and n ∈ n0, 1, 2, . . . }, 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, P). Answer the following questions.
(a) Let A and B be two disjoint subsets of Ω . Write down the smallest
σ-algebra containing A and B .
(b) Let X1 , X2 , . . . be independent random variables on (Ω , e, P) with Xn ~ Ber(1/n). Does Xn a-s-.. 0 as n - o? Prove or disprove this.
(c) Let e1 , e2 , . . . , en be a collection of σ-algebras on Ω . Show that the collection r defined by r = gk(n)=1ek is also a σ-algebra of subsets of Ω .
2023-05-19