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

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

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

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.

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?

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.

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

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 

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 

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 

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

Remedies available in case of IPR infringement?

How will autonomy and financial independence help an IPO, considering that it has to function within the policy framework given by the national government? Why cannot an IPO be completely free of government's control?.

What is defensive mechanism of protection of TK? Give one example of defensive mechanism of protection of TK?

How does the Fence Post approach differ from the pith and Marrow approach I determining the scope of the claims in a patent infringement case?

What do you understand by Prior Informed Consent? When this consent is required to be obtained and for what purpose?

Explain counterfeiting and piracy in detail. What are the economic and social consequences of counterfeiting and piracy?

Write a note on WIPO Internet domain name dispute resolution.

Discuss various stages involved in the processing of an application for acquisition of a patent for an invention. Discuss the scope if any for bringing down the cost for acquiring a patent.

What is Domain name? How do domain name dispute arise? Discuss in detail.