Statistics and probability

1. (a) Explain what is meant by the transition probability matrix of a homogeneous Markov chain. [5 marks] (b) Explain what is meant by the stationary distribution of a Markov chain? [5 marks] (c) A Markov chain has transition probability matrix, A, with entries ay, and stationary distribution . Write down an expression for the entries of the reverse Markov chain. [5 marks (d) Consider the following transition probability matrix of a homogo- neous Markov chain, with three states i,j and & (the TPM is in that order). If the stationary vector of the chain is (1/9, 2/9, 2/3), determine whether the Markov chain is reversible. 1 Ik 1- /0.2 0.2 0.6 0.1 0.6 0.3 K 0.1 0.1 0.8 [5 marks] (e) Let X1, X2, X, be a sequence of random variables resulting from the above Markov chain. If X = i and Xy = j what is the probability that X2 = k? [5 marks]Starting with the R Punkting for the simulation of the junegration - death process, modify it to include "birth's in addition to immigrations and death, Use your modified function to simulate a bug realisation Prom the birth-immigration- death process where the birth, death , and immigration rates are all one, starting from an initial condition of zero.Exercises 1 Find the transition function of the two-state birth and death process by solving the forward equation. N Consider a birth and death process having three states 0, 1, and 2, and birth and death rates such that lo = #2. Use the forward equation to find Poy(t), y = 0, 1, 2.Example I (Rosenberg [91]). Let f and g be PDFs with corresponding DFs F and G. Also, let (10) h(x, y) = f(x):(y)[1 + a(2F(x) - 1)(2G(y) - 1)]. where ja|