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STP801S - STOCHASTIC PROCESSES - 2ND OPP - JULY 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science honours in Applied Statistics
QUALIFICATION CODE: O8BSSH
LEVEL: 8
COURSE CODE: STP801S
COURSE NAME: STOCHASTIC PROCESSES
SESSION: July, 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
Prof. RAKESH KUMAR
MODERATOR:
Prof. PETER NJUHO
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
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Question 1. (Total Marks: 10)
(a) What do you mean by a Martingale. Discuss one example of martingale.
(5 Marks)
(b) A particle performs a random walk with absorbing barriers, say 0 and 4. Whenever it is at
position r (O<r<4), it moves to r+1 with probability p or to r-1 with probability gq, p+q=1. But
as soon as it reaches 0 or 4, it remains there. The movement of the particle forms a Markov
chain. Write the transition probability matrix of this Markov chain.
(5 marks)
Question 2. (Total marks: 10)
Classify the stochastic processes according to parameter space and state-space. Give at least
two examples of each type.
(10 marks)
Question 3. (Total marks: 10)
(a) What is the period of a Markov chain? Differentiate between periodic and aperiodic
Markov chains.
(5 marks)
(b) What is the nature of state 1 of the Markov chain whose transition probability matrix is
given below:
(5 marks)
0
1
2
0
01O
I
1/2 0 1/2
2 010
Question 4. (Total marks: 20)
(a) What is a Poisson process?
(5 marks)
(b) Let N(t) be a Poisson process with rate A > 0. Prove that the probability of n occurrences by
time t is given by
P(t)
=
(atyteAt
!
<#
=
12,3;
«as
(15 marks)
Question 5. (Total marks: 20)
(a) Show that the transition probability matrix along with the initial distribution completely
specifies the probability distribution of a discrete-time Markov chain.
(10 marks)
(b) Suppose that the probability of a dry day (state 0) following a rainy day (state 1) is 1/3 and
that probability of a rainy day following a dry day is 1/2. Develop a two-state transition
probability matrix of the Markov chain. Given that May 1, 2022 is a dry day, find the
probability that May 3, 2022 is a dry day.
(10 marks)
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Question 6. (Total marks: 10)
(a) Find the steady-state probabilities of the Markov chain whose one-step transition
probability matrix is given below:
(7 marks)
0
| 0 2/3
|
1/2 0 1/2
1/2 1/2 0
(b) State Ergodic theorem.
Q.7 (Total marks: 20)
(a) Derive Kolmogorov backward differential equations.
(b) Derive the steady-state probability distribution of birth-death process.
(3 marks)
(10 marks)
(10 marks)
END:OF QUESTION PAPER.......00.0msesxesnesnmameconsnenensen