ELECTRICAL NETWORKS AND REVERSIBLE MARKOV CHAINS
STEVEN P. LALLEY
The object of this part of the course is to show how the mathematics of finite electrical networks can be used in the study of reversible Markov chains on finite and countable state spaces.
CONDITIONAL EXPECTATION AND MARTINGALES
1. I NTRODUCTION
Martingales play a role in stochastic processes roughly similar to that played by conserved
quantities in dynamical systems. Unlike a conserved quantity in dynamics, which remains
constant in time,
MONOTONE COUPLING AND THE ISING MODEL
1. P ERFECT M ATCHING IN B IPARTITE G RAPHS
Definition 1. A bipartite graph is a graph G = (V, E) whose vertex set V can be partitioned
into two disjoint set VI , VO in such a way that every edge e E has one endpoint
CONTINUOUS TIME MARKOV CHAINS
STEVEN P. LALLEY
1. Introduction
Discrete-time Markov chains are useful in simulation, since updating algorithms are easier
to construct in discrete steps. They can also be useful as crude models of physical, biological,
and
LAPLACES METHOD, FOURIER ANALYSIS, AND RANDOM WALKS ON Zd
STEVEN P. LALLEY
1. L APLACE S METHOD OF ASYMPTOTIC EXPANSION
1.1. Stirlings formula. Laplaces approach to Stirlings formula is noteworthy first, because it
makes a direct connection with the Gauss
CONCENTRATION INEQUALITIES
1. A ZUMA -H OEFFDING I NEQUALITY
Chebyshevs inequality is one of the most useful gadgets in the toolbox of probability, but it
does have its drawbacks. For estimating probabilities of somewhat rare events, e.g., .05 or less, it
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 2
DUE MONDAY OCTOBER 10
Problem 1. Let cfw_N J be a Poisson point process in R2 with intensity function (x , y ) = h (y )
that depends only on the y coordinate. Let NR be the number of points in th
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 5
DUE WEDNESDAY NOVEMBER 2
Problem 1. Symmetries. Let P = (p (i , j ) be an irreducible transition probability matrix on
a finite state space Y . An automorphism (or symmetry) of the transition kern
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 4
DUE MONDAY OCTOBER 24
Problem 1. Residual Lifetime Distribution. Let cfw_Sn n0 be a discrete renewal process with
interoccurrence time distribution cfw_pk k 1 . Thus,
Sn =
n
X
Xj
j =1
where X 1 ,
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 7
DUE WEDNESDAY NOVEMBER 23
Problem 1. Birth-Death Chains. A birth-death chain on the nonnegative integers Z+ is an irreducible Markov chain on Z+ for which only transitions to nearest neighbors are
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 6
DUE WEDNESDAY NOVEMBER 16
In the following problems the sequence (Z n )n 0 is assumed to be a Galton-Watson process with
Z 0 = 1 and offspring distribution
P cfw_Z 1 = k = pk
Denote by (t ) =
P
k
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 1
DUE MONDAY OCTOBER 3
Problem 1. Let Nt and M t be independent Poisson counting processes with intensities , , respectively. Define to be the time of first occurrence in the process Nt , so that =
STATISTICS 313: STOCHASTIC PROCESSES II
HOMEWORK ASSIGNMENT 8
DUE WEDNESDAY NOVEMBER 30
In the following problems , W (t ) is a standard Wiener process, M (t ) is the maximum up to time t , and
for any a > 0, (a ) is the first time that W (t ) visits a ;
STATISTICS 312: STOCHASTIC PROCESSES
HOMEWORK ASSIGNMENT 3
DUE MONDAY OCTOBER 17
Problem 1. Let be a p : q random walk on the integers Z started at S0 = 0, i.e., let
Sn =
n
X
Xi
i =1
where the steps X i are independent, identically distributed with common
1. This problem is essentially the On Your Own part in Lab #6. Please complete the lab and
submit answers to the following questions.
(a) Make a histogram of price, which shows the population distribution of the sale price of all
2930 homes in the data se
3.4 Triathlon times, Part I. In triathlons, it is common for racers to be placed into age and gender groups.
Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men,
Ages 30 - 34 group while Mary competed in the Wome
1.16 Income and education in US counties. The scatterplot below shows the relationship between per
capita income (in thousands of dollars) and percent of population with a bachelors degree in 3,143
counties in the US in 2010
(b) Describe the relationship
2.10 Guessing on an exam. In a multiple choice exam, there are 5 questions and 4 choices for each
question (a, b, c, d). Nancy has not studied for the exam at all and decides to randomly guess the
answers. What is the probability that: (a) the first quest
1. The 2008 General Social Survey asked, What do you think is the ideal number of children for a family
to have? The 678 females who responded had a median of 2, mean of 3.22, and standard deviation of
1.99.
(a) What is the point estimate of the populatio
1.44 : Make-up exam. In a class of 25 students, 24 of them took an exam in class and 1 student took a
make-up exam the following day. The professor graded the first batch of 24 exams and found an average
score of 74 points with a standard deviation of 8.9
Lecture 1&2
Exploratory Data Analysis I Numerical Data
Yibi Huang
Department of Statistics
University of Chicago
Outline
In Lecture 1& 2, we cover mostly Section 1.2 & 1.6 in the text.
Data and Types of Variables (1.2)
Histograms (1.6.3)
Mean and Media
Lecture 7-8
Probability
Yibi Huang
Department of Statistics
University of Chicago
Outline
In Lecture 7-8, we cover mostly Section 2.1-2.3 in the text.
Probability and Events (2.1)
General Addition Rule (2.1.2-2.1.3)
The Complement Rule (2.1.4-2.1.5)
C
Lecture 5-6
Data Collection
Yibi Huang
Department of Statistics
University of Chicago
Outline
Lecture 5-6 covers mostly Section 1.1, 1.3, 1.4, 1.5 in the text.
Experiments (1.1, 1.3.4-1.3.5, 1.5)
Observational Studies (1.3.4-1.3.5, 1.4.1)
Sampling (1.3
Lecture 10-11
Continuous Distributions and
Normal Distributions
Yibi Huang
Department of Statistics
University of Chicago
Outline
In Lecture 10, we will cover Section 2.5 and 3.1 in the text.
Continuous distribution (2.5)
Normal distribution (3.1)
Pleas
Student Name (Print):
2017 Spring
(First Name)
STAT 22000
(Last Name)
Practice Midterm Exam
1. Do not sit directly next to another student.
2. Do not turn the page until told to do so.
3. You may use your calculator, and one letter-size formula sheet.
4.
Lecture 12
Binomial Distributions
Yibi Huang
Department of Statistics
University of Chicago
Outline
In Lecture 10, we will cover
Binomial distribution (3.4)
Please skip section 3.3 and 3.5.
1
Binomial distribution
Bernoulli Trials
A random trial having o
Lecture 16
General Framework of Hypothesis Testing
Yibi Huang
Department of Statistics
University of Chicago
Textbook Coverage
Lecture 16 covers section 1.8 (skip the simulation) and some of 4.3
in the text.
1
Case Study: Gender Discrimination
In 1972, a
Lecture 9
Random Variables
Yibi Huang
Department of Statistics
University of Chicago
Outline
In Lecture 9, we cover Section 2.4 in the text.
Random Variables
Expected Value
Standard Deviation
Linear combinations of random variables
1
Random variables
Lecture 14
Variability in Estimates and
Central Limit Theorem
Yibi Huang
Department of Statistics
University of Chicago
Outline
In Lecture 14, we will cover section 4.1 and 4.4 in the text, which
includes
Central Limit Theorem (CLT)
Sampling distributio
Events, Addition Rule, General Addition Rule
1. A card is selected at random from a deck of 52 poker cards. Let A be the event that the
selected card is a King, and let B be the event that the selected card is a Queen.
(a) Are events A and B disjoint?
(b)