STAT 333 Assignment 3
Due: Tuesday, July 14 at 2:30 pm (in class)
You may use your favourite math software for the matrix manipulation. Just include your output.
Type I Problems
1. #10, page 265 of the text, 9
th
edition.
2. #11, page 265 of the text, 9
th
edition.
3. #25, page 267 of the text, 9
th
edition.
4. #33, page 268 of the text, 9
th
edition.
Type II Problems
1.
Consider a sequence of repeated independent tosses of a
fair
coin, each toss resulting in H or
T. For each n = 1, 2, 3, . . . define X
n
= length of the run after the n
th
toss where a run is a
maximal sequence of like outcomes (i.e., all H or all T).
For example, if the sequence of outcomes looks like H H T H H H H T …
then X
1
= 1, X
2
= 2, X
3
= 1, X
4
= 1, X
5
= 2, X
6
= 3, X
7
= 4, X
8
= 1, etc.
a.
Model this as a Markov chain by writing down the state space S and transition matrix
P
.
b.
Prove that this chain is irreducible and find the period of the chain.
c.
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 Spring '08
 Chisholm
 Probability theory, Markov chain, discretetime Markov chain, H H T H H H H T

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