PI An B ) = o Yes , mutually exclusive as both events do not occure simultaneously 2 1 P( hove state n Heh) = o Yes , mutually exclusive 3) Pl tove state nc)= not mutually exclusive 41 Pl Hate state n A) = 1 2 . mutually exclusive option ( Al Q 1B ) are mutually exclusive Brent CS Scanned with CamScannerpublic class RightTriangle ( public int hypotenuse () [ private int base: return this -hypotenuse; private int height; private int hypotenuse; public void metBase (int newBans) public RightTriangle thin baco - nowBang [int bane; int height) get Hypotenuse ( ) this baoo - babe; this, height . height, getHypotempo () : public void setlleight (int newileig this. height . nowHeight ; set Hypotenuse ( ) public int base () { return chin , baser private void notHypotenuse () this. hypotenuse = (int) Math rou public int height ( Math. Dact (bage *base + height whoi return this . haight, and RightTriangle.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]University Department FTGradEnrollment Baylor University Statistics 26 Boston University Biostatistics 39 Brown University Biostatistics 21 Carnegie Mellon University Statistics 39 Case Western Reserve University Statistics 11 Colorado State University Statistics 14 Columbia University Biostatistics 64 Columbia University Statistics 196 Cornell University Statistics 78 Duke University Statistics 31 Emory University Biostatistics 58 Florida State University Statistics 47 George Mason University Statistics 10 George Washington University Statistics 9 Harvard University Biostatistics 70 Harvard University Statistics 67 Iowa State University Statistics 145 Johns Hopkins University Biostatistics 41 Kansas State University Statistics 44 Medical College of Georgia Biostatistics 11 Medical College of Wisconsin Biostatistics 7 Medical University of South Carolina Biostatistics 46 Michigan State University Statistics 81 New York University Statistics 6 North Carolina State University Statistics 163 North Dakota State University Statistics 25 Northwestern University Statistics 12 Ohio State University Statistics 101 Oklahoma State University Statistics 22 Oregon State University Statistics 30 Pennsylvania State University Statistics 75 Purdue University Statistics 85 Rice University Statistics 55 Rutgers University Statistics 111 Southern Methodist University Statistics 21 Stanford University Statistics 100 State University of New York - Buffalo Biostatistics 43 Temple University Statistics 40 Texas A&M University Statistics 101 University of Alabama - Birmingham Biostatistics 49 University of Arizona Statistics 3 University of California - Berkeley Biostatistics 36 University of California - Berkeley Statistics 58 University of California - Davis Statistics 34 University of California - Los Angeles Biostatistics 60 University of California - Los Angeles Statistics 72 University of California - Riverside Statistics 54 University of California - Santa Barbara Statistics 53 University of Chicago Statistics 109 University of Cincinnati Biostatistics 31 University of Connecticut Statistics 45 University of Florida Statistics 68 University of Georgia Statistics 59 University of Illinois Statistics 58 University of Iowa Biostatistics 35 University of Iowa Statistics 75 University of Kentucky Statistics 10 University of Massachusetts - Amherst Biostatistics 19 University of Michigan Biostatistics 117 University of Michigan Statistics 108 University of Minnesota Biostatistics 48 University of Minnesota Statistics 47 University of Missouri Statistics 58 University of Nebraska Statistics 44 University of North Carolina Biostatistics 118 University of North Carolina Statistics 78 University of Pennsylvania Statistics 23 University of Pittsburgh Statistics 32 University of Rochester Biostatistics 18 University of South Carolina Biostatistics 45 University of South Carolina Statistics 32 University of Texas - Houston Biostatistics 62A = det ( A - d I) = 0 6 4 - 1 3 (3 - 1) [ (9- 1) 13-2) - 16 ] A- dI = 16 - 6 [6 ( 3- 2) - 460] 6 4 4 + ( - 1) [ 24 + 9-] = 0 = ( 9 - 1 ) 1 3 - 1) - 16 ( 3 - ) - 36(3 - 1) - 24 - 33 = Q + 1 ( 1 - 62 + 9 ) ( 9 - 2) - 48 + 161 - 108 + 361 41 - 57 = 0. : 9 1 - 1 - 5 4 1 + 6d + 81- 9x - 48 + 16x - 108 + 364 + 1 -57= 0. - 13 + 15x2- 101 - 132 =0 123 - 151 +101 + 132 = Q solve the above cubic in calculator to get, A = 13.54 , = 3. 935 , A= - 2.476 = ) Now we find eigenvectors (lal Ax = 1x = 3 x1+ 6 *2- X3 = Ax, 6 x , + 9 * 2 + 4*3= *2 - * 1 + 4*2+ 3x3 = 1X3 - Solving we get (after normalizing to unit norm) X 1/ = 1-0. 4551 1 = 13.54 , Vy = ( x 2 1 = 1 - 0 . 84 60 X3 ) = - 0. 2778 2 1 = 3. 935: Similar procedure as above, gives 0. 5802 - 0. 0451 7 2= 3. 935 - D. 8132