M.Sc. (Semester II) (CBCS) Examination Oct/Nov-2017
Statistics
STOCHASTIC PROCESSES
Day Date: Wednesday, 22-11-2017 Max. Marks: 70
Time: 10.30 AM to 01.00 PM
Instructions: Q.1 and Q.2 are compulsory.
Attempt any three questions from Q. 3 to 7.
Figures to the right indicate full marks.
Q.1 Choose the correct alternative: 05
If is a Poisson process with parameter then var( N
t

For a non-null recurrent state the mean recurrent time is

1 0
All the entries of transition probability matrix are always
Positive Non-negative
Integer None of these
If for a symmetric random walk, probability of positive jump is 0.5,
then the random walk is called
Two sided Symmetric
Two fold None of these
If period of a state is two, then the state is called as
u niperiodic Aperiodic
Periodic None of these
Q.1 Fill in the blanks: 04
The TPM and specifies the Markov chain completely.
The collection of all possible states of the stochastic process is called as

Finite Markov chain contains at least one state.
For a persistent state the ultimate first return probability
A persistent state is also called as state.
Q.1 State true and false 04
TPM is always a square symmetric matrix.
State space of a Markov chain is countable set.
A counting process is a discrete state space stochastic process.
Persistency is a class property.
Q.2 Answer the following:- 06
Write a note on stochastic process.
Define and illustrate closed communicating class.
Write short notes on the following: 08
Counting process.
First entrance probability.
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SLR-MS-651
Q.3 Define and illustrate:- 07
State Space
Mean recurrent time
Doubly stochastic matrix
Transient State
Describe stationary distribution a Markov chain. Find the same for a Markov
chain, whose tpm is
1/3 2/3
1/9 8/9
07
Q.4 Show that a state is transient if and only if
∞

∞ 07
Prove that periodicity is a class property. 07
Q.5 Define first return probability and obtain it for every state of Markov chain
with state space and tpm P as

1/3 2/3 0
1/3 2/3 0
1/5 4/5 0
07
For the Markov chain with state space and tpm P as

1/7 1/7 5/7
1/5 2/5 2/5
1 0 0
find following probabilities.
07




It is given that chain is equally likely to start from any state of the state
space.
Q.6 Describe Yule-Furry process. Obtain the expression for for this
process.
07
Describe Poisson process. Obtain the distribution of inter-arrival times. 07
Q.7 Describe branching process. Derive the expression for expected number of
individuals at nth generation.
07
Write an algorithm to generate a realization from Markov process. 07