Stochastic Processes, Math 171, Spring 2015 - Homework 5
Due Friday May 1st
Read sections 1.5 and 1.6 in the book.
Solve problems 1.13, 1.14, 1.30, 1.34, 1.41, 1.49, 1.50 and 1.51 in the book.
Also solve the following.
Problem 1. For the problem 1 from th
Stochastic Processes, Math 171, Spring 2015 - Homework 3
Due Friday April 17th
Read section 1.3 in the book.
Solve problems 1.6 and 1.8 on pages 75/76 in the book.
Also solve the following.
Problem 1. Consider independent tosses of a fair die. Let Zn cfw_
Midterm 1, Math 171 - Spring 2015
Instructor: Toni Antunovi
c
c
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. If you have any questions raise your hand.
The correct nal answer alone is not sucient for full cre
Stochastic Processes, Math 171, Spring 2015 - Homework 3
Due Friday April 17th
Read section 1.3 in the book.
Solve problems 1.6 and 1.8 on pages 75/76 in the book.
Also solve the following.
Problem 1. Consider independent tosses of a fair die. Let Zn cfw_
Stochastic Processes, Math 171, Spring 2015 - Homework 6
Due Friday May 8th
Read sections 1.8 and 1.9 in the book.
Solve problems 1.56, 1.57, 1.59, 1.60, 1.62, 1.64, 1.65 and 1.69 in the book.
And problems below:
Problem 1. There are n cells labeled with
Stochastic Processes, Math 171, Spring 2015 - Homework 6
Due Friday May 8th
Read sections 1.8 and 1.9 in the book.
Solve problems 1.56, 1.57, 1.59, 1.60, 1.62, 1.64, 1.65 and 1.69 in the book.
And problems below:
Problem 1. There are n cells labeled with
Stochastic Processes, Math 171, Spring 2015 - Homework 1
Due Friday April 10th
Read section 1.1, 1.2 and 1.3 in the book.
Solve problems 1, 2, 3, 5 and 7 on page 75 in the book and 45 from page 84.
Also solve the following.
Problem 1. Let X1 , X2 , . . .
Midterm 2, Math 171, Spring 2015
Instructor: Toni Antunovi
c
c
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. If you have any questions raise your hand.
The correct nal answer alone is not sucient for full cred
Math 171, Spring 2015 - Homework 1
Due Friday April 3rd
Problem 1. Let X be a random variable with expectation 0 and variance 1.
Compute E[X 2 + X].
Problem 2. Let X be a Poisson random variable with parameter 1. Let Y be a Geometric
random variable with
Math 171, Spring 2015 - Homework 1
Due Friday April 3rd
Problem 1. Let X be a random variable with expectation 0 and variance 1.
Compute E[X 2 + X].
Solution: Since E(X 2 ) = var(X) + (EX)2 = 1 + 02 = 1 we have E(X 2 + X) = E(X 2 ) +
E(X) = 1 + 0 = 1.
Pro
Practice Final, Math 171 - Spring 2015
Printed name:
Signed name:
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Instructions:
Read problems very carefully. Please raise your hand if you have questions at any time.
The correct nal answer alone is not sucient for full credit - try
Practice Final, Math 171 - Spring 2015
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. Please raise your hand if you have questions at any time.
The correct nal answer alone is not sucient for full credit - try
Stochastic Processes, Math 171, Spring 2015 - Homework 3
Due Friday April 24th
Read section 1.4 in the book.
Solve problems 1.9, 1.10, 1.11 and 1.12 on pages 76/77 in the book.
Also solve the following.
Problem 1. Consider the Markov chain on states cfw_1
Practice Midterm 1, Math 171, Spring 2015
Instructor: Toni Antunovi
c
c
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. If you have any questions raise your hand.
The correct nal answer alone is not su cient for
171 Midterm 2 Solutions, Winter 20171
1. Question 1
Let = [0, 1]. Let P be the uniform probability law on . Let X : [0, 1] R be a random
variable such that X(t) = t3 for all t [0, 1]. Let
A = cfw_[0, 1/4), [1/4, 1/2), [1/2, 3/4), [3/4, 1]
Compute explicit
Math 171, Fall 2016, UCLA
Name:
Instructor: Steven Heilman
UCLA ID:
Date:
Signature:
.
(By signing here, I certify that I have taken this test while refraining from cheating.)
Final Exam
This exam contains 16 pages (including this cover page) and 10 probl
Math 171, Stochastic Processes, Winter 2017
Exterior Course Website: http:/www.math.ucla.edu/heilman/171w17.html
Prerequisite: Math 33A and Math 170A (or Statistics 100A). It is helpful, though not required,
to take Math 170B before this course or concurr
Math 171, Fall 2016, UCLA
Name:
Instructor: Steven Heilman
UCLA ID:
Date:
Signature:
.
(By signing here, I certify that I have taken this test while refraining from cheating.)
Mid-Term 1
This exam contains 8 pages (including this cover page) and 5 problem
171 Final Solutions, Fall 20161
1. Question 1
True/False
(a) Let cfw_N (s)s0 be a Poisson Process with parameter = 1. Then
N (4) N (3), N (3) N (2), N (2) N (1), N (1)
are all independent random variables.
TRUE. This is the independent increment property,
171 Midterm 1 Solutions, Fall 20161
1. Question 1
True/False
(a) Let be a universe. Let A1 , A2 ,S. . . . Then
FALSE. If A1 = and A2 = , then
i=1 Ai = , but = cfw_x : positive integers j, x
Aj .
(b) For any positive integers i, j, let aij be a real numb
Math 171, Fall 2016, UCLA
Name:
Instructor: Steven Heilman
UCLA ID:
Date:
Signature:
.
(By signing here, I certify that I have taken this test while refraining from cheating.)
Mid-Term 2
This exam contains 7 pages (including this cover page) and 4 problem
Stochastic Processes
Steven Heilman
Please provide complete and well-written solutions to the following exercises.
Due March 7, at the beginning of discussion section.
Homework 7
Exercise 1. Prove the following variant of the Optional Stopping Theorem. As
Stochastic Processes, Math 171, Spring 2015 - Homework 9
Due Friday May 29th
Read section 4.1, 4.2 and 4.3 in the book.
Solve problems 4.3, 4.4, 4.5 and 4.6 from the book, and the following problems.
Problem 1. Consider a continuous time Markov chain on a
Practice Midterm 1, Math 171, Spring 2015
Instructor: Toni Antunovi
c
c
Printed name:
Signed name:
Student ID number:
Instructions:
Read problems very carefully. If you have any questions raise your hand.
The correct nal answer alone is not sucient for
i
Essentials of Stochastic Processes
Rick Durrett
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Almost Final Version of the 2nd Edition, December, 2011
Copyright 2011, All rights reserved.
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Preface
Between
Mathematics 171 - Homework 3 : Due
Thursday, April 27, 2017.
Feel free to use the results in the homework problems (including previous
homework problems). In other words, you may utilize the problems to prove
each other as long as you can.
If not specifie
Mathematics 171 - Homework 1 : Due
Thursday, April 13, 2017.
Problems 1.1, 1.2, 1.3, 1.5, 1.7 on Page 62-63 in the textbook.
A1. Let A, B be events in a sample space. Let C1 , ., Cn be events such that
Ci Cj = for any i, j cfw_1, 2, ., n, and such that ni
Mathematics 171 - Homework 2 : Due
Thursday, April 20, 2017.
Problems 1.8 (a) (c), 1.9 (a) 1.10 (b). on Page 63-64 in the textbook.
Some of the following problems come from the first edition of the text
book.
B1. Consider the Markov chain with transition
Stochastic Processes, Math 171, Spring 2015 - Homework 8
Due Friday May 22nd
Solve the problems from Midterm 2 (you can nd it on CCLE week 8) and submit it as usual
homework.
1