Markov chain with State space

Memphis ences Daily AV Cale III Quiz # 4 Applications of Partial Derivatives "vel surface F(x, v.2) = ]x 2 + y'?'=4 goes through the point P late the gradient of F at No equations for the plane tangent to the surface and the line norm ent Plane: al Line: ler the bivariate function /(x y) = x - 3x+ y -4v+12. ate the derivative of / in the direction of u = (; at x I critical points for f, and use the second derivative te4. Consider a discrete-time Markov chain with the following probability transition matrix 0 0 P= I-T- VVO 0 0 1 Is it possible to choose values for ar and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. (b) The Markov chain is reducible. UNNN (c) The Markov chain has at least one transient state. (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4).2. A Markov chain with state space {1, 2, 3} has transition probability matrix 0.6 0.3 0.1 [Pa = 0.3 0.3 0.4 0.4 0.1 0.5 (a) Is this Markov chain irreducible? Is the Markov chain recurrent or transient? Explain your answers. (1)] What is the period of state 1? Hence deduce the period of the remaining states. Does this Markov chain have a limiting distribution? (0) Consider a general three-state Markov chain with transition matrix P11 P12 P13 1?: 1021 1022 p23 P31 P32 P33 Give an example of a specic set of probabilities pm; for which the Markov chain is not irreducible (there is no single right answer to this, of course l]