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Unformatted text preview: 1 Markov Chains Definition A Stochastic Process {X(t), t T} is a collection of random variables. That is, for each t T, X{t) is a random variable. The values of t is often interpreted as time (though not necessarily) and thus we say that X(t) is the state of the process at time t. T is called the index set . If T is a countable set, this stochastic process is said to be a discretetime process . If T is a subset of the real line. The stochastic process is said to be a continuoustime process . A state space is for a stochastic process is the set of all possible values that that the random variable X(t) can assume. This may also be countable or continuous. 2 Markov Chains Definition A Markov Chain is a Stochastic Process whose index set is the set of nonnegative integers n=0,1 ,2, and whose states space is countable or finite. We denote the process {X n , n=0,1,2,} and If X n =i , we say that the process is in state i at time n We define for all states i n1 , ,i , i, j and n 0 }  Pr{ } ,..., ,  Pr{ 1 1 1 1 i X j X i X i X i X j X P n n n n n n ij = = = = = = = = + + 3 Markov Chains Definition P ij are called the one step transition probabilities and represent the probability that a process that is in state i will transition (in one time unit) to state j And we define the one step transition matrix as = = = 1 , j ij ij P P 0,1,2,... i for = i2 i1 i0 12 11 10 02 01 00 P P P P P P P P P , P P ij Rows sum to 1 but not necessarily so for columns 4 Markov Chains Examples: Weather Forecasting If Pr{rain tomorrow  rain today} = If Pr{rain tomorrow  no rain today} = Define:  state 0 (Rain), state 1 (No Rain) X n is the state of weather on day n  = 1 1 P 5 Markov Chains Examples: Communications System Transmission of 0s and 1s each must pass through stages, let p denote the probability that a digit entering a stage is unchanged. X n is the state of the digit at time n  = p p p p P 1 1 6 Markov Chains Examples: Predicting Garys Mood Gary is cheerful ,C, soso, S, or glum , G Define: state 0 (C), state 1 (S) and state 2 (G) X n is the state of Garys mood on day n = 5 . 3 . 2 . 3 . 4 . 3 . 1 . 4 . 5 . P 7 Markov Chains Examples: Transforming to a Markov Chain P(Rain Tomorrow Rain Today, Rain Yesterday}= 0.7 P(Rain Tomorrow Rain Today, No Rain Yesterday}= 0.5 P(Rain Tomorrow No Rain Today, Rain Yesterday}= 0.4 P(Rain Tomorrow No Rain Today, No Rain Yesterday}= 0.2 Define: state 0: rained today and yesterday state 1: rained today but not yesterday state 2: rained yesterday but not today state 3: rained neither today nor yesterday X n is the state of weather 8 Markov Chains Examples: Transforming to a Markov Chain Define: state 0: rained today and yesterday...
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 Spring '11
 Mazzuchi

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