3.7 Exercises
:ation and Martingales
size of a population 3.7 Exercises
ndependent of what
.1 generation as inde 1- For f = {45:9}: Show that E[le] =
i Z 0 The DESprmg 2. Give the proof of Proposition 3.2 when X and Y are Jomtiy
' . Let '
eneration n + 1
Chapter 5 Markov Chains
[ark0v chain ever makes
starts in state i. Show
; with 3' then fr,- = 1.
of a Markov chain is a
acurrent and i does not
alk is the Markov chain
gers, and which has the
:19
1 p = 1/2, and transient
a chain is called the 1
walk.
atio
110
9.
10.
11.
12.
13.
14.
15.
Chapter 3 Conditional Expectation and Martingales
Suppose X1, X2, ., are independent and identically distributed
mean 0 random variables which each take value +1 with prob-
ability 1/2 and take value 1 with probability 1/2.
ation and Martingales
die of a population
idependent of what
. generation 71. inde
2 0. The offspring
neration n + 1. Let
ation 71. Assuming
ng of an individual,
at 2 0, is a martin-
Jrollary implies that
plies, when m < 1,
for all n sufciently
generation
6.10 Problems 121
formula (6.12) implies
E(S)
ll
[U] lljlngiyydm
1
[2.110(9) dy/U II$II2A($)dm-
Given this naive physical model and a constant Mm), passage to spheri
cal coordinates shows that it is possible for the three-dimensional integral
U |33|2A(2
Your name:
STATISTICS 204
FALL 2007
FINAL
This is a 3-hour in-class exam. There are 7 problems: do as many problems
as you can. You are not expected to do them all. Most questions do not
require long calculations. Students are allowed a calculator and not
1
Introduction
These problems are meant to be practice problems for you to see if you have understood the
material reasonably well. They are neither exhaustive (e.g. Diusions, continuous time
branching processes etc are not covered) nor are they meant to
74
Chapter 2 Stein's Method and Central Limit Theorems
2.7 Exercises
. If X N Poisson(a) and Y ~ Poisson(b), with b > a, use coupling
to show that _>_st X.
. Suppose a particle starts at position 5 on a number line and at
each time period the particle m