STA 447, 2006: Stochastic Processes
Exercises for week 1
Not for credit; do not hand in!
1.) (Durrett, p. 63, ex. 1.2) Five white balls and ve black balls are distributed
in two urns in such a way that each urn contains ve balls. At each step we draw one
STA 447, 2006: Stochastic Processes
Exercises for weeks 8 and 9
Not for credit; do not hand in!
1.) (Durrett, p. 176 , ex. 5.8) Let Sn = X1 + +Xn where the Xi are independent
2
with EXi = 0 and var(Xi ) = 2 . (a) Show that Mn = Sn n 2 is a martingale. (b)
STA 447, 2006: Stochastic Processes
Exercises for week 5
Not for credit; do not hand in!
1.) (Durrett, p. 64, ex.s 1.9, 1.10) Find the stationary distributions for the
Markov chains on S = cfw_1, 2, 3 with transition matrices:
0.5 0.4 0.1
0.5 0.4 0.1
a.)
STA 447, 2006: Stochastic Processes
Exercises for week 6
Not for credit; do not hand in!
1.) (Durrett, p. 69, ex. 1.41) Reecting random walk on the line. Consider points
0, 1 , 2 and 3, marked on a line. Let Xn be a Markov chain that moves to the right
wi
STA 447, 2006: Stochastic Processes
Exercises for week 4
Not for credit; do not hand in!
1.) (Lawler, p. 35, ex. 1.3,1.4) Consider a Markov chain Xn on state space
S = cfw_1, 2, 3. For each of
0.4
P = 0.6
0.2
the transition matrices
0.2 0.4 0.4
0.2 0.4
an
STA 447, 2006: Stochastic Processes
Exercises for week 2
Not for credit; do not hand in!
1.) (Grimmett & Stirzaker, p. 226, ex. 3) Classify the states of the Markov
chains with the following transition matrices:
a.)
1 2p
2p
0
p
1 2p
p
0
2p
1 2p
P =
b.)
0
Assignment2
STA447
Instructions: The assignment is due in class on April 7 at 6:10PM. No late assignments will
be accepted. All questions of equal value unless otherwise noted.
1. (i) Let X1 , X2 , . . . be iid with mean 0 and variance 2 . Set Sn = X1 + +
STA 447, 2006: Stochastic Processes
Exercises for week 7
Not for credit; do not hand in!
Exercises 3 and 4 from last weeks practice set will be recycled, as youre only now beginning
with martingales. Parts of exercises 1 and 4 require use of the optional
STA 447 Lecture Notes
Mark Koudstaal
January 27, 2011
1
1/25/2011
1.1
Tail Events, Kolmogorov 0-1
If we have any collection of -algebras cfw_Gn n>0 (where Gn F and (, F, P ) is our
probability space) then we may form the tail -algebra of this collection
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Name:
Student Number:
STA 447 / 2006 (circle one), Winter 2012, Mid-Term Test
(February 16, 2012. Duration: 60 minutes. Questions: 2. Pages: 3. Total points: 36.)
Notes: You should fully explain your answers! Continue on the back if necessary. NO
AIDS ALL
STA 447H1S STA 2006H1S Stochastic Processes
Course information Winter 2013 University of Toronto
Description
This course is an introduction to Stochastic Processes for graduate and fourth-year
undergraduate students. It will cover the standard examples of
Assignmenti STA447
Note: Due Thursday, February17 at the beginning of the lecture. Please
try to keep a copy of the original . This assignment is worth 16% of
your final grade. Plagerism => 0 . No late assignments .
1. Let {Xt : t=0,1,.} be a Galton Wat
University of Toronto
Winter Semester 2015
STA 447H1 Stochastic Processe and
STA 2006H18 Applied Stochastic
QUIZ 1 January 22, 20 5
I'OCBSSBS
PRINT YOUR NAME:
YOUR STUDENT ID NUMBER:
Total points: 12.
Do not skip steps in your solutions or fail t desc
Topic 4. Evolution of Branching Populations.
1. GabonWatson and Fisher Models. Probability Generating Functions.
The first model we will consider is due to Galron and Watson. It was
introduced in their paper titled "On the probability of extinction of
fam
Topic 3. Some Models of Population Dynamics.
Here, we will illustrate the concept of conditional probability and the
formula of total probability considered in Topics 1 and 2, reSpectively, by
studying several relatively simple models of the population dy