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 denote the average
number of plays it takes for the gamb
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
with parameters n and p. Use this result to nd the mean,
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( x 2) x 2
lim
+
1
x
2
3
4
5
2
3
4
5
-1
0.175
-2
0.15
0
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
6x
3 5 5 x
8 6 x
8 7 x 9 5 x
4x
8x
5x
9x
2x
7 4 8 x
Ev
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 14, 2008
ABOUT THE COVER:
THE WORK OF JESSE DOUGLAS ON M
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
13
1
1=
12
12
since P (A B ) 1.
For examples of attain
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 6.3.1 and Theorem 6.3.2 (b) are convenient
here.)
0
0 01
0
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. Homotopy and the Fundamental Group
1. Homotopy
2. The Funda