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STAT3004/STAT7304 Probability Models & Stochastic Processes Semester One Final Examinations, 2020
发布时间：2023-06-02
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School of Mathematics & Physics
EXAMINATION
Semester One Final Examinations, 2020
STAT3004/STAT7304 Probability Models & Stochastic Processes
(1) Consider a continuous time Markov chain {Xt}t≥0 on the states {0, 1} with generator matrix,
Q = 
] .
Consider also the discrete time process {Yn}
constructed via Yn = Xn∆ where ∆ > 0. Let p0 2 [0, 1] and assume P(X0 = 0) = p0 .
Compute expressions for each of these and provide the corresponding reasoning. Do not leave your answer in matrix form, but rather find explicit expressions:
(1a) limt!1 P(Xt = 1). (10pts)
(1b) P(Xt = 1) for finite t. (10pts)
(1c) P(Y2 = 1). (10pts)
(1d) limn!1 P(Xn+
Yn) assuming ∆ = 1. (10pts)
(2) Consider a beautiful tropical island with only 4 inhabitants (individuals) where each inhab- itant has their own coconut tree. The island also has a single shared telephone booth used for overseas calls. Each tree sheds coconuts according to its own Poisson process with rate λ > 0 and whenever this occurs and the individual is next to the tree, they go to the telephone booth to make an overseas call. They report over the phone: “I had a coconut drop!” .
If the telephone booth is busy, the individual queues up and waits until it is their turn to use the telephone. After an individual finishes their call they immediately return to their tree to continue with the same process. The duration of each call is exponentially distributed with rate µ > 0 and is independent of all other events.
(2a) Describe this as a continuous time birth and death process on the state {0, 1, 2, 3, 4} where the state describes the number of individuals near or in the telephone booth (either making a call or queueing). (10pts).
(2b) Assume λ = µ = 1. Determine,
tlim!1 
Z0t 1{Booth is free at time u}du,
where 1{·} is the indicator function. (10pts).
(2c) Relate this problem to an M/M/K/K queue. What is K? What is the birth rate? What is the death rate? (10pts).
(3) Consider a Markov chain {Xn}
on state space {1, 2, 3, 4, 5, 6, 7} with transition probability matrix,
|
6(6) 0
0 |
0.8 0.2 0.2 0 0 0 0 |
0 0 0 0.3 0 0 0 |
0 0 0.7 0 0.3 0 0 |
0 0 0 0.7 0 0 0 |
0 0 0 0 0.7 0 0.4 |
7 0 7(7) . 0 7(7)
0.6 |
(3a) Assume P(X0 = 7) = 1 and denote,
T = inf{n : Xn > Xn−1 }.
Determine a numerical value for P(T = 4). (10pts).
(3b) Determine numerical values for the elements of the 7 ⇥ 7 matrix M with entries,
Mij = nlim!1P(Xn = j | X0 = i).
A numerical answer is not enough, present how you obtain these results without brute force calculation. (20pts)
