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.
Let's tackle this step-by-step: --- ### Part 1: Chapman-Kolmogorov Equation #### Definition: The Chapman-Kolmogorov equation gives a relationship between *n-step* and *k-step* transition probabilities in a Markov chain. If $P^{(n)} = [p_{ij}^{(n)}]$ is the matrix of *n-step transition probabilities*, then the Chapman-Kolmogorov equation states: $$ p_{ij}^{(n+m)} = \sum_{k} p_{ik}^{(n)} \cdot p_{kj}^{(m)} $$ That is, the probability of transitioning from state $i$ to state $j$ in $n+m$ steps is the sum over all intermediate states $k$ of the probability of going from $i$ to $k$ in $n$ steps and then from $k$ to $j$ in $m$ steps. --- ### ________ _____ ____ ___ ___.
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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
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 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
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i)
ii)
Also find the expected duration in each case.
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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