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20092ee132B_1_hw6

# 20092ee132B_1_hw6 - in a trail 1 Show that N={N n N =0...

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UCLA Electrical Engineering Department EE132B HW Set #6 Professor Izhak Rubin Problem 1 Consider a Markov chain X = {X k , k = 0,1,…} with state space S = {a, b, c} and the transition matrix as follows: 1 1 1 2 4 4 2 1 0 3 3 3 2 0 5 5 P = . Compute 1. The steady state distribution ( 29 ( 29 ( 29 , , a b c π π π π =  , 2. ( 29 1 2 3 4 5 6 7 0 , , , , , , | P X b X c X a X c X a X c X b X c = = = = = = = = . 3. ( 29 2 | k k P X c X b + = = . Problem 2 Consider a discrete time Markov chain X with state space S = {a, b, c} and a transition matrix given as follows: 1 2 0 3 3 1 3 0 4 4 2 3 0 5 5 P = . The initial distribution for X is given by: ( 29 ( 29 ( 29 0 0 0 0 2 1 2 , , , , 5 5 5 a b c π π π π = = . Compute 1. The steady state distribution ( 29 ( 29 ( 29 , , a b c π π π π =  . 2. ( 29 1 3 4 6 0 , , , | P X b X a X c X b X a = = = = = 3. ( 29 1 2 3 , , P X b X b X a = = = . Problem 3

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Define N n to be the number of successes in n Bernoulli trials. Let p denote the probability of success
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Unformatted text preview: in a trail. 1. Show that N={N n : N =0, n=0,1,…} is a discrete time Markov chain. 2. Obtain the initial distribution and calculate the transition probabilities for N. Problem 4 Let {Y k : k=0,1,…} denote a sequence of independent and identically distributed (i.i.d.) discrete random variables. These random variables are governed by the probability distribution [p 1 , p 2 , …, p i , …], where p i = P{ Y k = i}, i=1,2,…. . Define 1 0 for 0, for 1,2,. .. n n k k n X Y n = = = = ∑ 1. Show that { } , 0,1,. .. n X X n = = is a Markov chain. 2. Compute the transition probability matrix for X in terms of {p i }....
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