LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – STATISTICS
|
FIFTH SEMESTER – April 2009
ST 5400 – APPLIED STOCHASTIC PROCESSES
Date & Time: 28/04/2009 / 1:00 – 4:00 Dept. No. Max. : 100 Marks
SECTION – A
Answer all the questions (10 x 2 = 20 )
- Define Stochastic Process with an example.
- What is the State Space of a Stochastic Process?
- Define Markov Process.
- Explain Independent Increments.
- Define Transition Probability Matrix.
- If P = is a Stochastic Matrix, fill up the missing entries in the Matrix.
- Define Accessibility of a State from another state.
- What is a Recurrent State?
- Define aperiodic Markov Chain.
- What is a Martingale?
SECTION – B
Answer any Five questions (5 x 8 = 40)
- State the classification of Stochastic Processes based on time and state Space. Give an example for each type.
- Describe One-dimensional Random Walk and write down its Transition Probability Matrix.
- Let { Xn, n ³ 0} be a Markov chain with three states 0,1,2 and with transition
probability matrix
and the initial distribution Pr{ X0 = i}= 1/3, i = 0,1,2
Find i) Pr{X1 = 1 ½ X0 = 2}, ii) Pr{X2 = 2 ½ X1 = 1}
iii) Pr{X2 = 2, X1 = 1 ½ X0 = 2}, iv) Pr{X2 = 2, X1 = 1, X0 = 2}
- Show that Communication is an equivalence relation.
- Consider the following Transition Probability Matrix. Using the necessary and sufficient condition for recurrence, examine the nature of all the three states.
- State any one property of Poisson Process.
- Classify the states of Markov Chain with Transition Probability Matrix
- State and stablish Chapman – Kolmogorov equations for a discrete time Markov chain.
SECTION – C
Answer any Two questions (2 x 20 = 40)
- Derive the distribution of X(t), is a Poisson Process. (20)
- a). State and prove a necessary and sufficient condition for a state to be Recurrent .
b). Explain the Two-dimensional Symmetric Random Walk. (10+10)
- (a) Let { Zi, i = 1,2…} be a sequence of independent identically distributed random variables with mean 1. Show that Xn = is a Martingale (8)
(b). Consider a Markov Chain with Transition Probability Matrix
Find the equivalence classes and compute the periodicities of all the four states. (12)
- a). Illustrate with an example Basic Limit Theorem of Markov chains..
b). Consider the following Transition Probability Matrix explaining seasonal changes on
successive day. (S – Sunny, C – Cloudy)
Today state
yesterday state
Compute the stationary probabilities and interpret the results. [10+10]