EE465-USC-Fall08-Assignment5_Solution

EE465-USC-Fall08-Assignment5_Solution - EE 465 Homework 5...

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Unformatted text preview: EE 465 : Homework 5 Solutions 1. As the hint suggests, define a state as what had happened in the last 3 days. Then, we have 8 states, namely { ( RRR ) , ( RRD ) , ( RDR ) , ( RDD ) , ( DRR ) , ( DRD ) , ( DDR ) , ( DDD ) } where D = dry and R = rain. For instance, ( DDR ) means that it is raining today, was dry yesterday and was dry the day before yesterday. 2. This is not a markov chain because transition probability is different for different n . This may be transformed to a markov chain by enlarging the state space to S = , 1 , 2 ,- ,- 1 ,- 2 where state i (- i ) signifies that the present value is i , and the present day is even (odd). 3. Proof by induction: For n = 1, P (1) = 1 2 + 1 2 (2 p 1) 1 2 1 2 (2 p 1 1 2 1 2 (2 p 1) 1 2 + 1 2 (2 p 1) = p 1 p 1 p p = P For n > 1, P ( n +1) = p 1 p 1 p p 1 2 + 1 2 (2 p 1) n 1 2 1 2 (2 p 1) n 1 2 1 2 (2 p 1) n 1 2 + 1 2 (2 p 1) n = a b b a (Since we know that the matrix is symmetric). a = p 1 2 + 1 2 (2 p 1) n +(1 p ) 1 2 1 2 (2 p 1) n = 1 2 ( p + p (2 p 1) n + 1 p (1 p )(2 p 1) n...
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EE465-USC-Fall08-Assignment5_Solution - EE 465 Homework 5...

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