if the coin is tossed independently 6 times, then the sequence of random variables {Xn, 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
For the coin tossed independently 6 times, define $X_n$ as the outcome on the $n$-th toss (for $n=1,2,\dots,6$). The parameter set (index set) is $T=\{1,2,3,4,5,6\}$, which is discrete. Each $X_n$ ___ ____ _________ ___ ___ _________ __________ __________.
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_______ __________ ___ __________ ___ _______ ___ ______.
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________ ____ __________ ___ ____.
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___ ____ _________ ______ __________ _____ _________.
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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
if the coin is tossed independently 6 times, then the sequence of random variables {Xn, 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
Define Martingales, Sub-Martingales and Super-Martingales with examples.
Describe the Branching processes with examples and explain some properties of generating functions of branching processes.
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?
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.
State whether the following statements are true or false and also give the reason in support of your answer: