Basic Probability Summer 2011
Assignment Two - additional problems
1) Consider the Gamblers Ruin Problem (page 17, Example 1.7.4 for p = 1/2
and page 74, Example 3.9.6 for case of general p.) Let Dk d
Basic Probability Summer 2011
Homework Number Four
Some problems on moment Generating functions and the
Central Limit Theorem
1) Compute the moment generating function of a binomial random variable
wi
Exam 2 Take Home
Name: _, _
(last)
Last Name
f(x)
(first)
Graph of f (x)
Graph of f (x)
Evaluate
0.5
lim
A
xe
x e2x
3
4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
1
+ ln x
x
5
-0.5
x 2
-1
-1.5
-2
1
1
BC
2
1
1
ln
Exam 1 Take Home
Last Name
AAm
AoAv
BCh
Co-D
EF
GK
LM
N
O
P
R
S
TY
Name: _, _
(last)
Differentiate
Differentiate
5 8 x 2 6 x
8 7 3 x
2 6 x 2 7 x
4 1 8 x
6 4 x 9 8 x
6 2 3 x
2 8 x 5 6 x
4 2 8 x
9 9 x
MATHEMATICAL PERSPECTIVES
BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 45, Number 2, April 2008, Pages 293302
S 0273-0979(08)01192-0
Article electronically published on February 1
Basic Probability: Problem Set One
Summer 2011
1.3.1 We have A B B P (A B ) P (B ) = 1 .
3
We also have from the inclusion-exclusion principle that
P (A B ) = P (A) + P (B ) P (A B )
13
=
P (A B )
12
Basic Probability Summer 2011
Assignment Five Addendum
Some problems on Markov Chains
1) For each of the following Markov chains, classify all the states as recurrent
or transient. (Note: Denition 6.3
Algebraic Topology
Len Evens
Rob Thompson
Northwestern University
City University of New York
Contents
Chapter 1. Introduction
1. Introduction
2. Point Set Topology, Brief Review
5
5
7
Chapter 2. Homo