Chapter 11
Three- (and higher-) way ANOVA
There is no problem handling models with more than two factors. If all factors are xed,
then the usual linear model theory applies. If some are, or all, facto
STAT 430
2GR, 2UG
1.
Spring 2016
A. Stepanov
Examples for 05/02/2016
Let B ( t ) be a (standard) Brownian motion. For fixed a, b R, consider the
following stochastic differential equation
dX( t ) =
b
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Martingale:
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Examples for 04/20/2016
discrete time
E( Xn + 1 | X0, X1, , Xn ) = Xn,
n 0.
continuous time
E [ X ( t ) | cfw_ X ( r ), 0 r s ] = X ( s ),
From
S
STAT 430
2GR, 2UG
1.
Examples for 04/11/2016
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A. Stepanov
Let X ( t ) be an Arithmetic Brownian motion with a drift coefficient = 0.3 and
diffusion coefficient = 0.4. Find the probability P
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The state space cfw_ 1, 2, 3, 4, 5, 6 2 is a square with
36 = 6 6 cells. A drunk person starting from ( 1, 1 )
tries to reach a house at ( 6, 6 ). The person moves one
cell up or on
STAT 430
2GR, 2UG
Examples for 01/25/2016
Let A S. The hitting time T A of A is defined by
Spring 2016
A. Stepanov
T A = min cfw_ n > 0 : X n A .
T A is the first (positive) time the Markov chain hits
STAT 430
2GR, 2UG
1.
Spring 2016
A. Stepanov
Examples for 03/14/2016
The Department of Statistics has two photocopy machines. The time to breakdown for
each machine has an exponential distribution wit
STAT 430
2GR, 2UG
Spring 2016
A. Stepanov
Examples for 04/18/2016
Brownian motion:
1)
W ( 0 ) = 0.
2)
If 0 s 1 t 1 s 2 t 2 , W ( t 1 ) W ( s 1 ) and W ( t 2 ) W ( s 2 ) are independent.
3)
W ( t ) W (
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Examples for 01/19/2017 (1)
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A. Stepanov
Example 1:
A rat runs through the following maze:
1
2
3
4
freedom
The rat starts in a given cell, and at each step, it moves to a
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2GR, 2UG
Spring 2016
A. Stepanov
Examples for 02/08/2016
Birth and death Markov chain:
qx
P ( x, y ) = r x
p
x
q0 = 0
y = x 1
y=x
y = x +1
0xd
pd = 0
OR
qx
P ( x, y ) = r x
p
x
= 1.
x0
Chapter 6
One-way ANOVA
This chapter will look more closely at the one-way ANOVA model. The model has g groups,
and Ni observations in group i, so that there are n = N1 + + Ng observations overall.
Fo
Chapter 10
Mixed Models
A mixed model is one with some xed eects and some random eects. The most basic
is the randomized block design, exemplied by the example on hot dogs in (9.1), where
the people a
Chapter 9
Random eects
So far, we have been concerned with xed eects, even though that term has yet to
be uttered. By xed, we mean the actual levels in the row or columns are of interest in
themselves
Some General Stat 448 Homework Suggestions/Tips
(many borrowed from Marias tips and used with her permission)
1. Read the entire homework assignment carefully, and make sure you do what each step of t
Homework 2
Due: Tuesday June 27 at 11:59pm
See general homework tips and submit your files via the course website.
The exercises are all based on data sets that came with the text book A Handbook of S
Guessing the Width of a Room Exercises
group=1
group=2
1) From the histograms, we might guess that the measures guessed in feet (group 2) tend to be a bit lower than
those made in meters in general an
Homework 1
Due: Tuesday June 20 at 11:59pm
See general homework tips and submit your files via the course website.
For all exercises, use the wine.data file and the code in HW1Data.sas in the course w
STAT 430
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Example 1:
P =
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Examples for 02/03/2016
Printer
broken
paper jam
need toner
working
broken
1
0
0
0
paper jam
0.09
0.16
0
0.75
need toner
0.03
0
0.12
0.85
worki
STAT 430
2GR, 2UG
Spring 2016
A. Stepanov
Examples for 02/12/2016
Birth and death Markov chain:
qx
P ( x, y ) = r x
p
x
q0 = 0
y = x 1
y=x
y = x +1
0xd
pd = 0
OR
qx
P ( x, y ) = r x
p
x
q0 = 0
y
STAT 430
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1.
Spring 2016
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Examples for 01/22/2016
The winter weather in Central Illinois is not kind there are never two nice
days in a row. If there is a nice day, the next day wou
STAT 430
2Gr, 2UG
1.
Spring 2016
A. Stepanov
Examples for 02/29/2015
Let T 1 , T 2 , , T k be independent Exponential random variables.
1
Suppose E ( T i ) =
, i = 1, 2, , k.
i
t
That is, f T ( t ) =
STAT 430
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Examples for 02/10/2016
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If ( x ), x S, are nonnegative numbers summing to one, and if
( x ) P ( x, y )
= ( y ),
y S,
xS
then is called a stationary distribut
STAT 430
2GR, 2UG
Spring 2016
A. Stepanov
Examples for 02/05/2016
Birth and death Markov chain:
qx
P ( x, y ) = r x
p
x
q0 = 0
y = x 1
y=x
y = x +1
0xd
pd = 0
0
1
2
3
4
d2
d1
d
0
0
0
0
0
0
0
0
0
0
STAT 430
2GR, 2UG
0f.
Examples for 03/04/2016
Spring 2016
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The state space cfw_ 1, 2, , 8 2 is a square with
64 = 8 8 cells. A drunk person starting from ( 1, 1 )
tries to reach a house at
STAT 430
2GR, 2UG
Spring 2016
A. Stepanov
Examples for 03/09/2016
Let X 1 ( t ) and X 2 ( t ) be independent Poisson processes with parameters 1 and 2 ,
respectively. Consider the processes X ( t ) =
STAT 430
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Examples for 02/15/2016
Let A S. The hitting time T A of A is defined by
Spring 2016
A. Stepanov
T A = min cfw_ n > 0 : X n A .
T A is the first (positive) time the Markov chain hits