What is the probability that you are in the upper

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Unformatted text preview: hat you are in the upper class (state 3) but your children are lower class (state 1)? Solution. Intuitively, the Markov property implies that starting from state 2 the probability of jumping to 3 and then to 1 is given by p(2, 3)p(3, 1) 8 CHAPTER 1. MARKOV CHAINS To get this conclusion from the definitions, we note that using the definition of conditional probability, P (X2 = 1, X1 = 3, X0 = 2) P (X0 = 2) P (X2 = 1, X1 = 3, X0 = 2) P (X1 = 3, X0 = 2) = · P (X1 = 3, X0 = 2) P (X0 = 2) = P (X2 = 1|X1 = 3, X0 = 2) · P (X1 = 3|X0 = 2) P (X2 = 1, X1 = 3|X0 = 2) = By the Markov property (1.1) the last expression is P (X2 = 1|X1 = 3) · P (X1 = 3|X0 = 2) = p(2, 3)p(3, 1) Moving on to the real question: Q2. What is the probability your children are lower class (1) given your parents were middle class (2)? Solution. To do this we simply have to consider the three possible states for your class and use the solution of the previous problem. P (X2 = 1|X0 = 2) = 3 X k=1 P (X2 = 1, X1 = k |X0 = 2) = 3 X p(2, k )p(k, 1) k=1 = (.3)(.7) + (.5)(.3) + (.2)(....
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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