STAT3004 Probability Models & Stochastic Processes Semester 1, 2022
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STAT 3004: Probability Models and Stochastic Processes
Final Exam, Semester 1, 2022
1. [8 Marks] Let (Sn , n = 0, 1, 2, . . . ) be a branching process with S0 = 1 and with offspring distribution having two offspring with probability
s, one offspring with probability r, and zero offspring with probability 1 - r - s, for r, s > 0 and r + s ≤ 1.
(a) Determine the probability generating function (pgf) of X, namely g(z) = EzX for z e [0, 1].
(b) Determine explicit expressions for the mean and variance of Sn as
a function of r and s, and determine through r and s when the process is sub-critical, critical, and super-critical.
(c) Determine the probability of ultimate extinction, denoted by η , as a function of r and s.
2. [6 Marks] Let X = (Xn , n = 0, 1, . . . ) be a Markov chain with state- space E = {0, 1, 2}, initial distribution π (0) = (1, 0, 0), and one-step
|
transition matrix
P = ╱
|
0.8 0.2 0 |
002、 1 │ . |
(a) Identify the communicating classes of the chain and classify the states of the chain.
(b) Determine the expected number of steps until the chain first enters state 2.
3. [8 Marks] A space ship is in search of a solar system with a habitable planet, which appears in space according to a homogeneous spatial Poisson process with rate of 3 solar systems with a habitable planet per thousand cubic light-years.
(a) What is the probability that the space ship finds n solar systems
with habitable planets within a radius of r thousand light-years?
(b) What is the expected number and variance of the number of solar
systems with habitable planets found by the space ship within a radius of r thousand light-years?
(c) For 0 ≤ s ≤ r , m e {0, 1, . . . , n}, and n e {0, 1, 2, . . . }, deter- mine the probability of m solar systems with habitable planets being within radius s thousand light-years, given n solar systems with habitable planets are within radius r thousand light-years. Identify this as a known distribution.
4. [8 Marks] Consider two machines maintained by two machine repair robots. Each machine has an exponentially distributed lifetime with mean 365 days (i.e., before it fails). Each 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. If one machine has failed, both robots work on that machine independently of each-other. If both machines have failed, one robot works on each machine. Both machine repair robots take an exponentially distributed amount of time, with mean 10 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 π(0) , and the Q-matrix Q. (b) Draw the corresponding transition rate diagram.
(c) What is the long-run probability that both machines are under repair?
(d) Suppose one machine has failed and is under repair. What is the probability that the machine repair robots fix the machine before the other machine fails?
5. [8 Marks] Consider an abstract probability space (Ω , F, P). Answer the following questions.
(a) Let B c A be two subsets of Ω . Write down the smallest σ-algebra containing A and B .
(b) Let X1 , X2 , . . . be independent random variables on (Ω , F, P) with Xn ~ Ber(1/n). Does Xn -
0 as n → o? Prove or disprove this.
(c) Let F1 and F2 be σ-algebras on Ω . Is g defined by g = F1 n F2 also a σ-algebra of subsets of Ω? Prove or disprove this statement.
2023-06-02