Markov chain.

5. A Markov chain {X,, n 2 0} with state space { 1, 2, 3, 4, 5 } has the transition probability matrix given below. DO P = Ooouo ooouo OHOOO HOO OOH (a) Draw the transition diagram for this Markov chain. (b) Is this Markov chain irreducible? Explain. (c) Determine the period of each state in this Markov chain.2. (15 pts) Consider a Markov chain { Xn } with state space S = {0, 1, 2} and transition matrix and transition matrix P = O ON/H HNIH O (1) Let the mapping f : S - S satisfy f(0) = 0 and f(2) = 1 and assume that f(1) + f(2). If Yn = f(Xn), then when is { Yn } a Markov chain? Is { Yn } always a Markov chain? In other words, are functions of Markov chains always Markov chains?View extends JPane/ and Implements Observer. RightTriangle to display is provided as constructor argument. Triangle View 1 Base 100 Height 50 Hypotenuse 1128. (10 points) (The Weak Law of Large Numbers) In order to estimate f, the true fraction of smokers in a large population, Alvin selects n people at random. His estimator M,, is obtained by dividing S, the number of smokers in his sample, by N, i.e., Mn = Sn. Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that P(IMn - 51 2 () 56, where e and o are some prespecified tolerances. Determine how the value of n recommended by the Chebyshev inequality changes in the following cases. The best guarantee will be when P(IMn - f1 2 () - Ane? (a) (5 points) The value of e is reduced to half its original value. (b) (5 points) If the error probability o is to be reduced to 6/2, while keeping e the same