M.Sc. (Semester II) (CBCS) Examination Oct/Nov-2017
Statistics
STOCHASTIC PROCESSES
Day Date: Monday, 20-11-2017 Max. Marks: 70
Time: 10:30 AM to 01.00 PM
Instructions: Q. No. and Q. No are compulsory.
Attempt any three from Q. No. to Q. No.
Figures to the right indicate full marks.
Q.1 Choose correct alternative: 05
For a null recurrent state the mean recurrent time is
∞
1 0
Addition of two Poisson processes is
A Poisson Process
May or may not be Poisson process
A Bessel process
Geometric process
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
For a doubly stochastic matrix,
Every row sum is one Every column sum is one
Both and None of these
If is a poisson process, then the inter-arrival times follow
Poisson distribution Exponential distribution
Geometric distribution None of these
Fill in the blanks. 05
If period of a state is one, then the state is called as
Finite Markov chain contains at least one state.
If is a Poisson process with parameter then
The initial distribution and specifies the Markov chain completely.
If for a state 1 then the state i is called state.
State the following sentence are True or False: 04
For any state


Poisson process is a counting process.
All the entries in a particular row of a TPM may be zero.
Persistency is a class property.
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SLR-MS-645
Q.2 Answer the following: 06
Write a note on stochastic process.
Define and illustrate 'lead relation.
Write short notes on the following: 08
Ergodic state.
Decomposition of a Poisson process.
Q.3 Define and illustrate: 07
Period of a state Persistent state
Absorbing state Transient state
Define mean recurrence time of a state. Find the same for both the states
of a Markov chain, whose tpm is
1/2 1/2
1/9 8/9
07
Q.4 Show that a state Í′ is transient if and only if
∞ ∞

. 07
Prove that transientness is a class property. 07
Q.5

1/3 2/3 0
1/3 2/3 0
1/5 4/5 0
Define first return probability and obtain it for every state of Markov chain
with state space and tpm P as
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

2


1



It is given that chain is equally likely to start from any state of the state
space.
Q.6 Obtain the expressions for for for Birth and Death process. 07
Verify the states of random walk model for persistency as well as for
periodicity.
07
Q.7 Describe branching process. Derive the expression for expected number of
individuals at generation.
07
If is a Poisson process, then for obtain the distribution of . 07