M.Sc. (Semester II) (CBCS) Examination Nov/Dec-2018
Statistics
STOCHASTIC PROCESSES
Time: 2½ Hours Max. Marks: 70
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 Select the most correct answer. 05
Suppose be a markov chain, then state j is transient iff

∞
∞

1
1
Let be a MC with state space 2 and tpm
0.3 0.3 0.4
0 0.2 0.8
0 0.8 0.2
Which of the following is true?
State 0 and 1 are communicative
All states are recurrent
State 1 is transient
State 1 and 2 are recurrent
Number of visitors to a restaurant on highway by time can be modeled
as
Markov chain Poisson process
Birth and death process Branching process
In a Branching process if then
n nm
mn none of the above
Suppose that customers arrive at a Bank according to a Poisson with
mean rate of 2 per unit. Expected number of arrivals in an interval of 8
minutes is
2 8
4 16
Fill in the blanks: 05
Temperature at time t is an example of time,
state space stochastic process.
State i is said to accessible from state j if
If state k is null persistent then
n
kk p →
If a state is not a persistent state, then it is called as
BGW branching process becomes extinct with probability one if the
offspring mean is
Page 2 of 2
SLR-VR-479
State whether following statements are true or false: 04
In an irreducible chain, all states are aperiodic.
In birth and death process, if and for all then the
process is called as Poisson process.
If 0 is Poisson process, then waiting time for kth event, 1
has gamma distribution.
Yule-Furry process is a particular process of Pure Birth process.
Q.2 Define: 06
Markov chain
One step transition probability matrix
Period of a state
Write short notes on: 08
Branching process
Chapman Kolmogorov equations
Q.3 Consider the problem of sending a binary message, 0 or through a
signal channel consisting of several stages, where transmission through
each stage is subject to a fixed probability of error Suppose that
0 is the signal that is sent and let be the signal that is received at the
stage. Assume that is a Markov chain.
07
Determine the transition probability matrix of
Determine the probability that no error occurs up to stages n 2
Determine the probability that a correct signal is received at stage 2
In an irreducible Markov chain, prove that all states are of same type. 07
Q.4 Discuss the classification of states of Markov chain in detail. 07
A professor continually gives exams to her students. She can give three
possible types of exams, and her class is graded as either having done
well or badly. Let pi denote the probability that the class does well on a
type exam, and suppose that 0.3, 0.6, and 0.9. If the class
done well on an exam, then the next exam is equally likely to be any of the
three types. If the class does badly, then the next exam is always type 1.
What proportion of exams are type
07
Q.5 Define Poisson process giving its two definitions. Also establish
equivalence between these two definitions.
07
Describe birth and death process and obtain its Kolmogorov differential
equations.
07
Q.6 Define random walk model. Also classify its states. 07
Show that the probability of ultimate extinction of a BGW branching
process is the smallest root of the equation P s where is the pgf
of the offspring distribution of the process.
07
Q.7 Discus Gambler's ruin problem in detail. 07
Write an algorithm for the simulation of Branching process 0 with
1. Hence write an procedure to estimate 20 and