Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 1024

Winter 2014
Statistical Science 1024
Chapter 1
Picturing Distributions with Graphs
Figure 1.1
A spreadsheet displaying data from the American Community Survey, for Example 1.1.
Available at US Census Bureau website.
Individuals and Variables
Any set of data contains
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 1024

Winter 2014
Statistical Science 1024
Chapter 4
Scatterplots and Correlation
Flipping 10 coins
Different from flipping 1000 coins
Z score
Facilitate comparison
Netflix prize
Scatter Plot
Scatterplot
4
The most useful graph for displaying the relationship between
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 1024

Winter 2014
Statistical Science 1024
Chapter 2
Describing Distributions with
Numbers
Henry Cavendish (1731 1810)
British scientist
carried out wideranging
scientific studies that
included
chemistry
electricity
physics
astronomy
Cavendish Laboratory for
physics
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 1024

Winter 2014
Statistical Science 1024
Chapter 5
Regression
Origin of the Term Regression
Francis Galton, 1886,
Regression towards
mediocrity in hereditary
stature. Journal of the
Anthropological Institute,
15: 246 263
See JSTOR under UWO
library databases
Example
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 1024

Winter 2014
Statistical Science 1024
Chapter 3
The Normal Distribution
From histogram to density curve
From histogram to density curve
More on Density Curves
a density curve describes the theoretical
pattern or distribution of the data that can
be obtained. The de
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
# Markov Chain Monte Carlo
# WALS 1  October 21, 2016
# Metropolis Hastings Algorithm
# Implimentation of the Metropolis Hastings Algorithm to generate P matrix
# Inputs are an initial 'Q' matrix and the 'b' vector
MH < function(Q,b)cfw_
# identify the
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Section 4.9
Background
Suppose that you are interested not so much in terms
of a particular random variable X, but rather some
function h(X) of that random variable.
Suppose further that it is the expected outcome
E[h(X)]; denote this expected outcome b
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Section 4.9
A direct way to craft the
probabilities , via the , s
Our decision to make this a timereversible Markov
chain provides us with the means to determine the
, s. Since time reversibility means (i) = (j) ,
we will use these facts to construct
Western University (Ontario)  Also known as University of Western Ontario
Regression
STATISTICS 3859

Fall 2016
Model Adequacy Checking
Regression Model:
y = X +
Assumptions:
1. The relationship between y and the predictors is linear.
2. The noise term has zero mean.
3. All s have the same variance 2 .
4. The s are uncorrelated between observations.
5. The s a
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Section 4.2
The ChapmanKolmogorov equations
Let () = + = = for all 0 where
is a Markov chain. Then for all m, > 0; , 0,
(+) =
()
()
=0
We will prove this result in class.
If we look at this equation closely, it resembles the
equation for m
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Section 4.3
Example 4.18 The Random Walk
Consider a Markov chain defined on all the integers:
An individual starts at at an arbirtrary integer (which
we might as well consider to be = 0).
In each step of the Markov chain, with probability
0 <
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Sections 4.3, 4.4
The reason to classify states
As we have seen through many examples, it is possible
for there to be a number of possible outcomes when
one investigates as n gets large.
It is possible that there is a limiting matrix, all of
wh
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Section 4.1
A Stochastic Process:
A stochastic process is defined in your text as
a collection of random variables (p. 84).
A better definition is that a stochastic process is this:
A stochastic process X(t) is a collection (or family) of
rando
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Sections 4.7
4.7 Branching Processes
A branching process is one model to study whether a
family line survives or eventually dies out.
One starts with a single individual, who produces j 0
offspring with probability . Each of these offspring in
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Section 4.8
4.8 Timereversible Markov Chains
One can look at a ergodic Markov chain that has been
operating forever in reversed time.
It is the case that this reversedtime process is also a
Markov chain, as we can show in class. Let
= 1 = =
Western University (Ontario)  Also known as University of Western Ontario
Markov chains with application
STATISTICS 4654

Fall 2016
Textbook Sections 4.5, 4.6
4.5 The Gamblers Ruin Problem
Consider a modification of the Random Walk:
The state i represents the present wealth of an
individual, who gambles $1 at each play of a game.
With probability 0 < < 1 the gambler wins $1; with
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 1024

Winter 2014
Statistics for Sociology
Chapter 1
Descriptive Statistics
Relevant in different situations
o When the researcher needs to summarize or describe the distribution of a single
variable (univariate descriptive statistics)
o When the researcher wishes to desc
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 2035

Spring 2016
Chapter 5 Discrete Random Variables and
Probability Distributions
Definition
A random variable is a variable which
assigns a numerical value to every outcome
of a random experiment
There are two types of random variables:
discrete and continuous
1. Discre
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 2035

Spring 2016
Bayes Rule (section 4.8)
Sometimes you need to determine posterior
probabilities based on your prior
probabilities
That is, you may want to determine P(AB)
given that you know P(A) and P(BA).
P(AB) =
P( A B)
P( B)
=
P ( A) P ( B  A)
P( B)
The above is
Western University (Ontario)  Also known as University of Western Ontario
STATISTICS 2035

Spring 2016
(Skip Counting Rules (Section 4.3) for now)
Conditional Probability and Independent
Events (sections 4.6/4.7)
Sometimes probabilities need to be
reevaluated as additional information
becomes available.
Example 4.6
Suppose in example 4.1, someone has
inspe