Homework 4, Math 3225, Fall 2010
November 22, 2010
1. In this problem I will walk you through a proof that the sample variance
is unbiased. Let
n
2
2
i=1 (Xi X )
S=
n1
denote the sample variance, where we assume X1 , ., Xn are i.i.d.
a. First show that E(
Homework 1, Math 3225
September 10, 2010
1. Recall that events A1 , ., Ak are independent if for every non-empty
subset S of cfw_1, ., k we have
P(sS As ) =
P(As ).
(1)
sS
As a consequence of this, it turns out that this implies, and is equivalent
to, th
Study Sheet for Math 3225, Exam 1, Fall 2010
October 8, 2010
1. Know basic denitions and results from set theory; for example, know
the two forms of de Morgans law, know distributive rule of intersection
and union (whichs says A(B C ) = (AB )(AC ) and A(B
Syllabus for Math 3225, Honors Probability
and Statistics, Fall 2010
August 24, 2010
Instructor: Ernie Croot
Oce: 103 Skiles
Oce Hours: TBA
Place and Date of Classroom: MWF, 1:05 - 1:55 in Skiles 271.
Book: Probability and Random Processes, 3rd Edition, b
1
Introductory Comments
First, I would like to point out that I got this material from two sources: The first was a page from Paul Graham's website at www.paulgraham.com/ffb.html, and the second was a paper by I. Androutsopoulos, J. Koutsias, K. V. Chandr
Notes on Bernoulli and Binomial random
variables
October 1, 2010
1
1.1
Expectation and Variance
Denitions
I suppose it is a good time to talk about expectation and variance, since they
will be needed in our discussion on Bernoulli and Binomial random vari
The Birthday Paradox
September 18, 2003
1
Two Birthdays the Same
It turns out that there is at least a 50% chance that in any random sample of 23 people, two of them will have the same birthday. By "birthday", I mean you don't include the year; so, an exa
Applications of the Central Limit Theorem
November 2, 2005
1
Introduction
First, we state the central limit theorem
Theorem 1 Suppose that X1 , X2 , . is an innite sequence of independent,
identically distributed random variables with common mean = E(X1 )
Notes on the Chi-Squared Distribution
October 19, 2005
1
Introduction
Recall the denition of the chi-squared random variable with k degrees of
freedom is given as
2
2
2 = X 1 + + X k ,
where the Xi s are all independent and have N (0, 1) distributions. Al
Notes on random variables, density functions,
and measures
September 29, 2010
1
Probability density functions and probability measures
We dened a probability density function (pdf for short) as a function f :
R [0, ) satisfying
f (x)dx = 1.
R
(Of course,
Notes on the second moment method, Erds
o
multiplication tables
January 25, 2011
1
Erds multiplication table theorem
o
Suppose we form the N N multiplication table, containing all the N 2
products ab, where 1 a, b N . Not all these products will be distin
Some facts about expectation
October 22, 2010
1
Expectation identities
There are certain useful identities concerning the expectation operator that
I neglected to mention early on in the course. Now is as good a time as any
to talk about them. Here they a
Homework 2, Math 3225, Fall 2010
October 7, 2010
1. Suppose that X and Y are independent normal random variables with
mean and variance 2 . Show that X + Y is also a normal random
variable for arbitrary constants and , not both 0. Determine its
mean and v
Homework 3, Math 3225, Fall 2010
October 24, 2010
1. Find an example of a pair of random variables X and Y such that
E(XY ) = E(X )E(Y ).
2. Compute (and provide proof) of the moment generating function for
the discrete random variable having mass functio
Study Sheet for Math 3225, Final Exam
December 13, 2010
This test will NOT be open note; however, I will give you some
selected notes from the course to use during the nal.
1. Know basic denitions and results from set theory; for example, know
the two for
Math 3225 Final Exam, Fall 2005
December 13, 2010
1. Dene the following
a. A sigma algebra.
b. The Kolgomorov axioms of probability.
c. The Law of Large Numbers.
d. The Central Limit Theorem.
2. Bob wishes to transmit one bit of information across a noisy
Notes on hypothesis testing
November 21, 2010
1
Null and alternate hypotheses
In scientic research one most often plays o some hypotheses against certain
others, and then one performs an experiment to decide whether to reject or
not reject certain of thes
Notes 1 for Honors Probability and Statistics
Ernie Croot
August 26, 2003
1
1.1
Set Theory
Basic Denitions
In mathematics a set is a collection of elements or objects. We also allow S
to have no objects, and we call this special kind of set the empty set,
Notes 2 for Honors Probability and Statistics
Ernie Croot
August 24, 2010
1
Examples of -algebras and Probability Measures
So far, the only examples of -algebras we have seen are ones where the
sample space is nite. Let us begin with a FALSE example of a
Notes 3 for Honors Probability and Statistics
Ernie Croot
September 12, 2003
1
Measure Theory and Integration
Now that we have a measure on B (R), namely the Borel-Lebesgue measure,
we can try to see what measure it assigns to certain sets. The main tool
Markov Chain, part 2
December 12, 2010
1
The gamblers ruin problem
Consider the following problem.
Problem. Suppose that a gambler starts playing a game with an initial
$B bank roll. The game proceeds in turns, where at the end of each turn
the gambler ei
Markov Chains, part I
December 8, 2010
1
Introduction
A Markov Chain is a sequence of random variables X0 , X1 , ., where each
Xi S , such that
P(Xi+1 = si+1 | Xi = si , Xi1 = si1 , ., X0 = s0 ) = P(Xi+1 = si+1 | Xi = si );
that is, the value of the next
Some practice problems involving Markov
Chains
December 20, 2010
1. Show that if T is a Markov chain such that there is a positive probability
of transitioning from any vertex to any other vertex in a single time
step (note: we are not saying that there i
Maximum likelihood estimators and least
squares
November 11, 2010
1
Maximum likelihood estimators
A maximum likelihood estimate for some hidden parameter (or parameters,
plural) of some probability distribution is a number computed from an
i.i.d. sample X
Some notes on the Poisson distribution
Ernie Croot
October 7, 2010
1
Introduction
The Poisson distribution is one of the most important that we will encounter
in this course it is right up there with the normal distribution. Yet, because
of time limitatio
Notes on Poisson Processes
October 6, 2010
1
Introduction
Depending on the book (or website) you read, a Poisson Process can have
many dierent denitions. For me, the key axioms dening it are as follows:
First, we x a time interval [0, T ], and a certain p
Noes on the correlation coecient
November 15, 2010
1
Introduction
There are two types of correlation coecients: the sample correlation coefcient, and the random variable analogue. Here, we will analyze and prove
the properties of the random variable versi
Estimation of Parameters and Statistical
Sampling
November 6, 2010
1
Introduction
Here we consider two types of statistical sampling problems, one is just for
pedagogical purposes, the other is directly applicable to real problems. These
two problems are: