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Unformatted text preview: if the ﬁrst urn contains i black balls
and m i white balls while the second contains b i black balls and m b + i
white balls. Each trial consists of choosing a ball at random from each urn and
exchanging the two. Let Xn be the state of the system after n exchanges have
been made. Xn is a Markov chain. (a) Compute its transition probability. (b)
Verify that the stationary distribution is given by
✓ ◆✓
◆✓◆
b 2m b
2m
⇡ (i) =
i
mi
m 71 1.12. EXERCISES (c) Can you give a simple intuitive explanation why the formula in (b) gives
the right answer?
1.47. Library chain. On each request the ith of n possible books is the one
chosen with probability pi . To make it quicker to ﬁnd the book the next time,
the librarian moves the book to the left end of the shelf. Deﬁne the state at
any time to be the sequence of books we see as we examine the shelf from left
to right. Since all the books are distinct this list is a permutation of the set
{1, 2, . . . n}, i.e., each number is listed exactly once. Show that
⇡ (i1 , . . . , in ) = pi1 · 1 pi2
·
pi1 1 pi3
···
pi1 pi2
1 pi1 pin
· · · pin...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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