MATH 170A, PROBABILITY THEORY, WINTER 2016
STEVEN HEILMAN
Abstract. These notes closely follow the book of Bertsekas and Tsitsiklis, available here.
Contents
1. Introduction
2. Sets and Probabilities
2.1. Sets
2.2. Probabilistic Models
2.3. Conditional Pr
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Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due February 18, in the discussion section.
Homework 6
Exercise 1. Suppose there are ten separate bins. You first randomly place a sphere
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due March 10, in the discussion section.
Homework 8
Exercise 1. Let X be a continuous random variable with distribution function fX (x) =
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due February 11, in the discussion section.
Homework 5
Exercise 1. Let X be a discrete random variable with finite variance. Let t R. Con
170A Midterm 1 Solutions1
1. Question 1
Label the following statements as TRUE or FALSE. If the statement is true, explain your
reasoning. If the statement is false, provide a counterexample and explain your reasoning.
In the following statements, let A,
170A Midterm 2 Solutions1
1. Question 1
(a) The number of ways to make an ordered list of k elements of the set cfw_1, 2, . . . , n is
n!/(n k)! = n(n 1) (n k + 1).
TRUE. Proposition 2.66 in the notes.
(b) Let n be a positive integer. Let be a discrete sa
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due January 14th, in the discussion section.
Homework 1
Exercise 1. Let A, B, C be sets in a universe . Using the definitions of intersec
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due January 28, in the discussion section.
Homework 3
Exercise 1. Let = [0, 1][0, 1] so that R2 . Define a probability law P so that, for
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due January 21, in the discussion section.
Homework 2
Exercise 1. Two fair coins are flipped. It is given that at least one of the coins
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due February 4, in the discussion section.
Homework 4
Exercise 1. The Wheel of Fortune involves the repeated spinning of a wheel with 72
Probability Theory 170A
Steven Heilman
Please provide complete and wellwritten solutions to the following exercises.
Due January 7th, in the discussion section.
(This Review Assignment will be collected but not be graded.)
Preliminary Review Assignment
E
Some useful formulas.
PMF of Bernouli (p):
pX (1) = p, pX (0) = 1 p.
Expectation is p, variance is p(1 p).
PMF of Bin(n, p):
n k
p (1 p)1k , for k = 0, 1, 2, , n.
pX (k) =
k
Expectation is np. Variance is np(1 p).
PMF of Geo(p):
pX (k) = (1 p)k1 p, f
Linear Algebra, Math 170A, Fall 2015, Lec 1  Course Info
Instructor: Martin Tassy, 5117 Math Sciences Building, mtassy@math.ucla.edu,
Instructor Oce Hours: MWF 1011am in 6909 Science Building
Lectures: Monday, Wednesday and Friday, 33:50pm in 6229 Math
Probability Theory, Math 170A, Fall 2015  Homework 9
From the textbook solve the problems 18, 20, 22, 25, 34 from the Chapter 3.
Solve the problems 10, 12, 15, 17 from the Chapter 3 additional exercises at
http:/www.athenasc.com/probsupp.html
And also t
Probability Theory, Math 170A,  Homework 8, DUE MONDAY NOVEMBER 30
From the textbook solve the problems 6, 7, 11 and 15 at the end of the Chapter 3.
Solve the problems 3, 6, 7, 8 and 14 from the Chapter 3 additional exercises at
http:/www.athenasc.com/pr
Probability Theory, Math 170A,  Homework 7
From the textbook solve the problems 1 and 2 at the end of the Chapter 3.
And also the problems below:
Problem 1. If X and Y are independent random variables and E(X) = 0 show that
E(X Y )2 ) = E(X + Y )2 ).
Doe
Probability Theory, Math 170A,  Homework 5
From the textbook solve the problems 16, 22, 24 at the end of the Chapter 2.
Solve the problems 5 and 13 from the Chapter 2 additional exercises at:
http:/www.athenasc.com/probsupp.html
And also the problems be
Probability Theory, Math 170a, Homework 2
Solve the problems 49,50,51,52,53,56,58,60 from the Chapter 1
And also the problems below:
Problem 1. Assume that 0 m n. Give a combinatorial proof that
n
m
n
=
k=m
k1
.
m1
(Hint: how many melement subsets of cfw
Probability Theory, Math 170A, Fall 2015, Homework 4
From the textbook solve the problems 3 to 7 at the end of the Chapter 2.
And also the problems below:
Problem 1. In a certain soccer tournament you are playing once with each
of the other nine teams. In
Final practice, Math 170A  Fall 2015
Instructor: Martin Tassy
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Total:
100
Score
1. (a) (2 points) Let A and B be events such that P(A B) = P(A B) = 1/2. Find P(A).
(b)
Final practice, Math 170A  Fall 2015
Instructor: Martin Tassy
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Total:
100
Score
1. (a) (2 points) Let A and B be events such that P(A B) = P(A B) = 1/2. Find P(A).
Sol