Probability Theory, Math 170a, Homework 1
From the textbook solve the problems 2, 5-10 at the end of the Chapter 1.
And also the problems below:
Problem 1. Show that for any sets A and B
P(A B) P(A) P(A B).
Problem 2. You want to buy a car on a certain we
Mathematics Department, UCLA T. Richthammer
winter 09, nal Mar 02, 2009
Final: Math 170A Probability, Sec. 1
1. (6 pts) Consider a probability model with S = cfw_1, 2, 3, 4, P (cfw_1, 2) = 1 , P (cfw_1, 3) = 1 . 2 4 (a) Is it possible that P (cfw_4) = 0?
Math 170A Winter 2013
Homework 4
(1) Read sections 2.2, 2.3 and 2.4 from the book. Also: nish reading the excerpt from Silvers book.
(2) Solve problems 1 and 6 from the end of chapter II.
(3) Solve problems 1 through 5 from the Practice midterm (posted on
Mathematics Department, UCLA T. Richthammer
winter 09, sheet 5 Jan 30, 2009
Homework assignments: Math 170A Probability, Sec. 1
051. An apartment complex is equipped with an alarm system that is supposed to directly give alarm at a police station if there
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
And also the problems below:
Problem 1. Recall Problem form Hom
Probability Theory, Math 170a, Winter 2015, Toni Antunovi c
c
Homework 1
From the textbook solve the problems 2 and 10 at the end of the Chapter
1.
Solve the problems 1, 2, 3 and 5 from the Chapter 1 additional exercises at
http:/www.athenasc.com/prob-sup
Math 170A Fall 2013
Homework 7
Suggested reading:
Review Chapters I, II, Sections 3.1 and 3.2 from Chapter III and Sections 4.2 and 4.3 from Chapter
IV for the midterm.
The solutions to Problem 3 and 9 at the end of Chapter II.
Problems:
(1) Solve probl
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/prob-supp.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 m-element subsets of cfw
4.1 Introduction
CHAPTER 4
Probability
While the graphical and numerical methods of Chapters 2 and 3 provide us
with tools for summarizing data, probability theory, the subject of this chapter,
provides a foundation for developing statistical theory. Most
Mathematics Department, UCLA T. Richthammer
winter 09, sheet 1 Jan 02, 2009
Homework assignments: Math 170A Probability, Sec. 1
001. Let E1 , E2 , . . . be subsets of a universal set S . Draw Venn diagrams for the following sets: (a) E1 E2 E3 , (d) E1 Ans
Mathematics Department, UCLA T. Richthammer
winter 09, sheet 10 Mar 06, 2009
Homework assignments: Math 170A Probability, Sec. 1
120. Express the following probabilities in terms of the joint CDF F of X = (X1 , X2 ): (a) P (X1 a, X2 > b) Answer: (a) P (X1
Chapter 1 Problems
(1) Let S, T, U and S1 , S2 , . . . be sets. Prove that
c
(a) S (T U ) = (S T ) (S U ),
(b)
Si
c
Si .
=
i=1
i=1
(2) Before the early 1990s, a telephone area code in the US consisted of three digits, where the rst
was not 0 or 1, the sec
Topology/Metric Spaces
1
Before we begin
d(x, z) = |x z| = |x y + y z| =
|(x y) + (y z)| |x y| + |y z| =
d(x, y) + d(y, z)
Before we discuss topological spaces in their full generality, we will rst turn our attention to a special type of
On
the
plane
R2
Math 115, Fall 2012
Practice Final December , 2012
Total possible points: 220
1. (10 point) Prove (2 + 2)1/2 is irrational.
2. (30 points) Decide (with proof) whether the following series converge.
1.
2.
3.
1
n=2 n2 n
n
2n
1
(1)n cos n
3. (30) poins) Let
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
Chapter 3: Joint Distributions
1
Introduction
This chapter is concerned with the joint probability structure of two or more random variables dened
on the same sample space.
In ecological studies, counts of several species, modeled as random variables, ar
Midterm 1, Math 170a - Lec 3, Fall 2012
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 fu
Midterm 1 practice, Math 170a - Lec 3, Fall 2012
Instructor: Toni Antunovi
c
c
Name and student ID:
Question
Points
1
12
2
10
3
10
4
10
5
8
Total:
50
Score
1. (a) (6 points) If = A B , P(A B c ) = 0.6, P(Ac B ) = 0.2 nd the probabilities of A and B .
Solu
Math 170A Winter 2013
Homework 2
(1) Read sections 1.4, 1.5, 1.6 and the solutions to problems 13 and 47 at the end of the chapter.
(2) Solve the following problems from the end of chapter I: 17, 33, 40, 49 and 50.
(3) Two fair six sided dice are rolled,
Math 170A Winter 2013
Homework 3
(1) Read sections 1.5 and 1.6 (again!), and the scans from Nate Silvers book.
(2) Solve the following problems from the end of chapter I: 52, 53, 54, 49 and 55.
(3) How many ways can 8 people be seated in a row, if
(a) The
Linear Algebra, Math 170A, Fall 2015, Lec 1 - Course Info
Instructor: Martin Tassy, 5117 Math Sciences Building, [email protected],
Instructor Oce Hours: MWF 10-11am in 6909 Science Building
Lectures: Monday, Wednesday and Friday, 3-3:50pm 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/prob-supp.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, 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
Probability Theory, Math 170A, - Homework 6
From the textbook solve the problems 25,26,31,32 at the end of the Chapter 2.
Solve the problems 12,14, 15 ,16 from the Chapter 2 additional exercises at:
http:/www.athenasc.com/prob-supp.html
And also the probl
Math 170A - Homework 4 - Due 16 February 2017
For all questions in this homework, no justification needed; your final answers are enough.
1. Let X be a discrete random variable. Find EX if
(i)
P(X = 1.1) = 0.5, P(X = 0) = 0.3, P(X = 2.5) = 0.2,
(ii) X Ber