The random variables W1, W2,... are independent with common distribution k 1 2 3 4 Pr( W = k) 0.1 0.3 0.2 0.4 Let Xn max (W1,..., Wn) be the largest W observed to date. Determine the transition probability matrix for the Markov chain {Xn}.

what does the equation, d/dt Π = MΠ calculate for where Π is population vector describing the overall state probability distributions and M is a 4x4 transition rate matrix?

2. Let Xo, X₁,... be the Markov chain on state space {1,2,3,4} with transition matrix (1/2 1/2 0 0 1/7 0 3/7 3/7 1/3 1/3 1/3 0 0 2/3 1/6 1/6/ (a) Explain how you can tell this Markov chain has a limiting distribution and how you could compute it.

Let X(1), X(2), ..., X(n) be the order statistics of a set of n independent uniform (0, 1) random variables.Find the conditional distribution of X(n) given that X(1) = s(1), X(2) = s(2), ..., X(n-1)=s(n-1).

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