Question

Please answer this in specific way,thanks.

1. A Markov chain X = (X2) >0 with state space I = {A, B, C} has a one-step transition matrix P given by 70 2/3 1/3) P= 1/3 0 2/3 (1/6 1/3 1/2) (a) Find the eigenvalues 11, 12, 13 of P. (b) Deduce pn can be written as pn = 10 + XU, + Aug (n > 0) and determine the matrices U1, U2, U3 by using the equations n = 0,1,2. (c) Verify your general formula for pn when n = 3. (d) Show that, for all i, j el, P(X, = j|X, = i) +1; as n +00 for some r = (TA, B, c) which you should specify. Interpret intuitively. [3] (e) Calculate P(X3 = A|Xo = A) and P(X2 = C, X6 = A|Xo = B). (f) Justifying your steps, find an expression for | P(Xem = B; X+m+n = X = A; Ag =C) where l, m, n € {1,2,...}. What happens as m, n o ? [5] (g) Let T be an independent random time with P(T = k) = (1 – alak (k > 0), for some a € (0,1). Let Q = (0i.jijel where dis := P(X7 =j|X= i). Find Q explicitly in terms of q. Explain intuitively what happens as a →1. [5]

Answer 1

ustract 23Td Tow (a) P - frem /3 2/2 2md: 16 c erisiden: p- 21 0, X's-> yemvaluel T-33dentity matri 6 Muatipry 2nd row by ad -X. 5ustraet 5t Tow 익3 - -+) -to

-4) ]. > - [-11 + 12 .E ]+ -4) 12 -%-1 2人 12 + + 12 5 4 72 3 2. +312 12. + BA + X + 54 + x12 54 2 2 + 2/ 24 72 31+16- 2 子2 > + 72 + 24 24 2 - +X 4 =>-0 2-1- 0 => 2-2入+>-I-o -) 2>(>-リ +1 (>-1) =0 ()-り (2)+リー。 => () Now -U3 = 0, t5/6 4/6 - 1 -1 So, UisU2,レ3 →dedenminied,