Suppose that a Markov chain (X,)n>o has a stochastic matrix given by: 1/2 1/2 1/4 3/4 1/3 1/3 1/3 P = 1/6 2/3 1/6 2/3 1/3 1/6 5/6 1/2 0 0 0 1/2 (c) What is P4(H' < H')? (d) What is P2(H' < H')? (e) Suppose now that the Markov chain is initialised from 1, which is a uniform distribution over all states 1 to 7. What is Pi(H' < H")? (f) What is the probability that a Markov chain, which

Consider the following. В 3 { (20, 2, 4), (8, 1, 2), (24, 3, 7)}, в' %3D {(-5, 3, -9), (-2, 1, -3), (-6, 3, -8)}, [x]g = 2 (a) Find the transition matrix from B to B'. 20 8 24 1 3 p-1 4 (b) Find the transition matrix from B' to B. 20 8 24 2 1 P = 2 7 (c) Verify that the two transition matrices are inverses of each other. PP-1= (d) Find the coordinate matrix [x]B, given the coordinate matrix [x]gB'. [x]B = Need Help? Read It 00 J00 00 00

1. (a) lind autocovariance function and autocorrelation function of Poisson Process. (b) If X(1) = A cos ct+ Bsin et and Y) and B are uncorrelated random variables with mean 0 and variance 1. Examine whether {X(1)} and {Y(!)} are jointly widen -sense stationary or not. Bcoset - Asin ct. where c is a constant, A

(6) Let Q1 be an m x n matrix with orthonormal columns and Q2 be an n x p matrix with orthonormal columns. Show that Q1Q2 has orthonormal columns.

A square matrix is said to be doubly stochastic if itsentries are all nonnegative and the entries in each row andeach column sum to 1. For any ergodic, doubly stochasticmatrix, show that all states have the same steady-stateprobability.