Mathematics 136, Midterm 2008 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use three pages (2 sides each) of notes. You have 90 minutes to solve all three questions, each worth points as m
Stat219 / Math 136 - Stochastic Processes Homework Set 1, Fall 2008. Due: Wednesday, October 1 For questions on grading, see TBA 1. Exercise 1.1.3. Let (, F , IP) be a probability space and A, B, Ai events in F . Prove the following properties of IP. (a)
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Math136 / Stat219 Course Goals
Basic concepts and denitions of measure-theoretic
probability and stochastic processes
Properties of key stochastic processes and their
applications, especially Brownian motion
Key results and common techniques of proof P
Last Time
Introduction Measurable space Generated -elds Borel -eld
Todays lecture: Sections 1.11.2.2
MATH136/STAT219 Lecture 2, September 24, 2008 p. 1/14
Probability space
A probability space is a triple (, F , I ), where (, F ) is a P
measurable space
Stat/219 Math 136 - Stochastic Processes Notes on Markov Processes
1
Notes on Markov processes
The following notes expand on Proposition 6.1.17, and attempt to minimize confusion between the terms stationary and homogeneous.
1.1
Stochastic processes with
Stochastic Processes Amir Dembo (Revised by Kevin Ross) September 25, 2008
E-mail address : [email protected], [email protected] Department of Statistics, Stanford University, Stanford, CA 94305.
Contents
Preface Chapter 1. Probability, measur
Stat/219 Math 136 - Stochastic Processes Notes on Section 4.1.2
1
Independence, Uncorrelated-ness, and Something in Between
Suppose (X, Y ) is a random vector dened on some probability space (, F , IP) with IE(X 2 ) < , IE(Y 2 ) < . The following implicat
Stat 219 - Stochastic Processes Homework Set 2, Fall 2008, Due: October 8 See Victor Hu for questions on grading 1. Exercise 1.2.22. Write (, F , P ) for a random experiment whose outcome is a recording of the results of n independent rolls of a balanced
Mathematics 136, Midterm 2007 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use three pages (2 sides each) of notes. You have 90 minutes to solve all three questions, each worth points as m
Mathematics 136, Final Winter 2008 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use six pages (2 sides each) of notes. You have 180 minutes to solve all four questions, each worth points a
Mathematics 136, Final Winter 2007 Write your name and sign the Honor code in the blue books provided. This is a closed material exam, but you may use six pages (2 sides each) of notes. You have 180 minutes to solve all four questions, each worth points a
Math 136 - Stochastic Processes Homework Set 4, Winter 2008, Due: February 6 1. Exercise 2.3.16. Let Z = (X, Y ) be a uniformly chosen point on (0, 1)2 . That is, X and Y are independent random variables, each having the U (0, 1) measure of Example 1.1.11
Math 136 - Stochastic Processes Homework Set 5, Fall 2008, Due: October 29 See Victor Hu on questions on grading 1. Exercise 3.2.21. Consider the random variables Sk of Example 1.4.13. (a) Applying Proposition 3.2.6 verify that the corresponding character
Math 136 - Stochastic Processes Suggested Excercises, Autumn 2008 Questions? See Bo Shen. 1. Exercise 4.6.8. Suppose cfw_Zn is a branching process with P(N = 1) < 1 and Z0 = 1. Show that P( lim Zn = ) = 1 pex ,
n
rst in case m 1, then in case P(N = 0) =
Math136/Stat219 Fall 2008 Midterm Examination Friday, October 24, 2008, 11:00am - 12:30pm Write your name and sign the Honor code in the blue books provided. You have 90 minutes to solve all questions, each worth points as marked (maximum of 50). Complete
Math136/Stat219 Fall 2008 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. Y
Math136/Stat219 Fall 2008 Sample Final Examination Write your name and sign the Honor code in the blue books provided. You have 3 hours to solve all questions, each worth points as marked (maximum of 100). Complete reasoning is required for full credit. Y
Math136/Stat219 Fall 2008 Sample Midterm Write your name and sign the Honor code in the blue books provided. You have 90 minutes to solve all questions, each worth points as marked (maximum of 50). Complete reasoning is required for full credit. You may c
Math136/Stat219 Fall 2008 Sample Midterm
Write your name and sign the Honor code in the blue books provided.
You have 90 minutes to solve all questions, each worth points as marked (maximum of 50). Complete reasoning is required for full credit. You may c
Math 136 - Stochastic Processes Homework Set 3, Autumn 2008, Due: October 15 Questions? See Bo Shen. 1. Exercise 1.4.30. Use Monotone Convergence to show that
E(
n=1
Yn ) =
n=1
EYn ,
for any sequence of non-negative R.V. Yn . Deduce that if X 0 and An a