1) Consider the following Markov chain with 52!≈8×10^67 states. The states are the possible orderings of a standard 52-card deck. To run one step of the chain, pick 2 different cards from the deck, with all pairs equally likely, and swap the 2 cards. Is the stationary distribution of the chain uniform over the 52! states, i.e., is the stationary distribution 1/52!(1,1,…,1) ? Yes or no?
2)Alice and Bob are wandering around randomly, independently of each other, in a house with M rooms, labeled 1,2,…,M . Let d subscript i be the number of doors in room i (leading to other rooms, not leading outside). At each step, Alice moves to another room by choosing randomly which door to go through (with equal probabilities). Bob does the same, independently. The Markov chain that each of them follows is irreducible and aperiodic. Find the limit of the probability that Alice is in room i and Bob is in room j , as time goes to ∞ .
3)Find the probabilities of states 1 , 2 , and 4 in the stationary distribution of the Markov chain s shown below. The label to the left of an arrow gives the corresponding transition probability.
I'm not very educated in college mathz as I'm still in primary school.
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