IGNOU MSTE 12 SOLVED ASSIGNMENT

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MSTE 012 (January 2025 - July 2025) - ENGLISH

TUTOR MARKED ASSIGNMENT MSTE-012: STOCHASTIC PROCESSES

Course Code: MSTE-012

Assignment Code: MSTE-012/TMA/2025

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are true or false and also give the reason in support of your answer:

a) if the coin is tossed independently 6 times, then the sequence of random variables {X_n, n ≥ 1} is a stochastic process with discrete parameter space T and discrete state space S.

b) A Markov chain . has the transition probability matrix

Then the conditional probabilities 

c) Suppose that customers arrive at a facility according to a Poisson process having rate be the number of customers that have arrived up to time t. Then the probability 

d) In the classical gambler's ruin problem, with total stake a and gambler's stake k, andthe gambler's probability of winning at each play is p. Then the probability of ruin if given that 

e) The number of customers at time t waiting in the queue including the one being served, if any, is called arrival rate.

2.(a) State the Chapman-Kolmogorov equation and mention the relation between higher and lower transition probabilities, it establishes.

Given the following transition matrix of a Markov chain with three states 1, 2 and 3:

find the matrix of three-step transition probabilities and, hence, obtain the transition probability

A symmetric random walk starts at x = 0. Find the probabilities that the walk

(i) s at x = 0 after 10 steps;

(ii) s at x =1 after 5 steps;

(iii) s at x = –3 after 9 steps.

3. A fire and emergency rescue service receives calls for assistance at a rate of ten per day. Teams man the service in twelve hour shifts. Assume that requests for help form a Poisson process

(i) What is the probability that a team would receive six requests for help in a shift?

(ii) What is the probability that a team has no requests for assistance in a shift?

4. In the standard gambler’s ruin problem, with total stake ? and gambler’s stake ?, and the gambler’s probability of winning at each play is ?, calculate the probability of ruin in the following cases;

i) 

ii)

Also find the expected duration in each case.

5. Define the Ordinary, modified and equilibrium renewal processes with examples. Also, differentiate between modified and equilibrium renewal processes.

6. Define Martingales, Sub-Martingales and Super-Martingales with examples.

7. Describe the Branching processes with examples and explain some properties of generating functions of branching processes. 

8. Describe the finite and infinite M/M/1 queuing models with examples and their waiting time distribution

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