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cards is ⇡ N log N , while the variance is ⇡ N 2 k=1 1/k 2 . 75 1.12. EXERCISES 1.69. Algorthmic e ciency. The simplex method minimizes linear functions by
moving between extreme points of a polyhedral region so that each transition
decreases the objective function. Suppose there are n extreme points and they
are numbered in increasing order of their values. Consider the Markov chain in
which p(1, 1) = 1 and p(i, j ) = 1/i 1 for j < i. In words, when we leave j we
are equally likely to go to any of the extreme points with better value. (a) Use
(1.25) to show that for i > 1
Ei T1 = 1 + 1/2 + · · · + 1/(i 1) (b) Let Ij = 1 if the chain visits j on the way from n to 1. Show that for j < n
P (Ij = 1Ij +1 , . . . In ) = 1/j to get another proof of the result and conclude that I1 , . . . In 1 are independent. Inﬁnite State Space
1.70. General birth and death chains. The state space is {0, 1, 2, . . .} and the
transition probability has
p(x, x + 1) = px
p(x, x 1) = qx
p(x, x) = rx for x > 0
for x 0 while the other p(x, y ) = 0. Let Vy = min{n 0 : Xn = y }...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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