- G And H Please 1 (172.16 KiB) Viewed 12 times
G and H please
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answerhappygod
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G and H please
G and H please
(27 points) Let X = {Xn: n=0,1,2,...} be a discrete-time Markov chain (DTMC) with state space S = {1,2,3,4,5,6,7,8,9} and transition matrix 0 0 .8 0 1 0 0 0 .5 0 0 1 .2 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 P= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .2 .6 0 0 0 .4 2.4 0 0 0 0 0 .5.5 0 0 .5 0 .5 0 0 .5 5 0 .2 0 0 0 0 0 0 0 (a) (3 points) Draw the transition diagram. For each state, determine whether it is transient, null recurrent, or positive recurrent. (b) (3 points) For each recurrent state, determine its period. (c) (4 points) Is the stationary distribution unique? If so, why. If not, please find two stationary distributions. (d) (3 points) Starting from state 2, find the probability that the Markov chain reaches state 1 before reaching state 4. = = = (e) (3 points) Let T{1,4} be the number of steps needed for the DTMC to reach either state 1 or state 4. Find the following probabilities i. P{T{1,4} = 1|Xo = 2} ii. P{T{1,4} = 2|X0 = 2} iii. P{T{1,4} = 3|Xo = 2} (f) (3 points) Find E(T{1,47|Xo = 2), the expected number of steps needed for the DTMC to reach either state 1 or state 4, given that it starts from state 2. (g) (4 points) Find approximately i. P{X100 = 5|Xo = 6}, ii. P{X100 = 7|X0 = 8} ~, iii. P{X100 = 7|X, = 6} , iv. P{X100 = 1|X0 = 2} , (h) (4 points) Find approximately i. P{X101 = 5|X0 = 6}, ii. P{X101 = 7|X0 = 8} , iii. P{X101 = 7|X0 = 6} ~, iv. P{X101 = 1|X0 = 2} ,
(27 points) Let X = {Xn: n=0,1,2,...} be a discrete-time Markov chain (DTMC) with state space S = {1,2,3,4,5,6,7,8,9} and transition matrix 0 0 .8 0 1 0 0 0 .5 0 0 1 .2 0 0 .5 0 0 0 0 0 0 0 0 0 0 0 0 P= 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .2 .6 0 0 0 .4 2.4 0 0 0 0 0 .5.5 0 0 .5 0 .5 0 0 .5 5 0 .2 0 0 0 0 0 0 0 (a) (3 points) Draw the transition diagram. For each state, determine whether it is transient, null recurrent, or positive recurrent. (b) (3 points) For each recurrent state, determine its period. (c) (4 points) Is the stationary distribution unique? If so, why. If not, please find two stationary distributions. (d) (3 points) Starting from state 2, find the probability that the Markov chain reaches state 1 before reaching state 4. = = = (e) (3 points) Let T{1,4} be the number of steps needed for the DTMC to reach either state 1 or state 4. Find the following probabilities i. P{T{1,4} = 1|Xo = 2} ii. P{T{1,4} = 2|X0 = 2} iii. P{T{1,4} = 3|Xo = 2} (f) (3 points) Find E(T{1,47|Xo = 2), the expected number of steps needed for the DTMC to reach either state 1 or state 4, given that it starts from state 2. (g) (4 points) Find approximately i. P{X100 = 5|Xo = 6}, ii. P{X100 = 7|X0 = 8} ~, iii. P{X100 = 7|X, = 6} , iv. P{X100 = 1|X0 = 2} , (h) (4 points) Find approximately i. P{X101 = 5|X0 = 6}, ii. P{X101 = 7|X0 = 8} , iii. P{X101 = 7|X0 = 6} ~, iv. P{X101 = 1|X0 = 2} ,