This preview shows page 1. Sign up to view the full content.
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 = 1X1 = 3, X0 = 2) · P (X1 = 3X0 = 2) P (X2 = 1, X1 = 3X0 = 2) = By the Markov property (1.1) the last expression is
P (X2 = 1X1 = 3) · P (X1 = 3X0 = 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 = 1X0 = 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)(....
View
Full
Document
This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

Click to edit the document details