Midterm 2, Math 170B — Lec. 1, Fall 2015
Instructor: Pierre-Frangois Rodriguez
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Printed name:
Signed name:
Student ID number:
Instructions:
0 Read the following problems very carefully.
o The correct ﬁnal answer alone is not sufﬁcient for fu
Probability Theory, Math 170b, Winter 2013, Toni Antunovi c
c
Homework 1
From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end
of the Chapter 4.
And also the problems below:
Problem 1. Give examples of (not independent) random variab
Probability Theory, Math 170b, Winter 2013, Toni Antunovi c
c
Homework 1
From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end
of the Chapter 4.
And also the problems below:
Problem 1. Give examples of (not independent) random variab
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4
c
c
solutions
From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4.
Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7
addi
Math 170B Practice Exam 1
1. (15 points) Let X, Y be two random variables, with var(X) = 1, var(Y ) = 4, cov(X, Y ) = 1.
Find var(X + 2Y ).
2. (20 points) Suppose X has the Cauchy density
fX (x) =
1
, x R.
(1 + x2 )
Find the probability density function o
Midterm 1, Math 170B v Lee. 1, Fall 2015
Instructor: Pierre—Francois Rodriguez
Printed name:
Signed name:
Student 'ID number:
Instructions:
0 Read the following problems very carefully.
o The correct ﬁnal answer alone is not sufﬁcient for full credi
Probability Theory, Math 170b, Spring 2015, Toni Antunovi
c
c
Homework 3
Due Friday, April 17th
From the textbook solve the problems 17, 18 and 19 at the end of the
Chapter 4.
From the books supplementary problems, solve problem 30 in Chapter 4
(see http:
Midterm 2, Math 170b - Lec 1, Winter 2013
Instructor: Toni Antunovi
c
c
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 n
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4
c
c
From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4.
Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7
additional exe
Midterm 3, Math 170b - Lec 1, Winter 2013
Instructor: Toni Antunovi
c
c
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 n
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6
c
c
Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the problems below:
Problem 1. Denote points P0 ,
Midterm 2, Math 170b - Lec 1, Winter 2013
Instructor: Toni Antunovi
c
c
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 n
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3
c
c
From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4.
Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at
http:/www.athenasc.com
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 2
c
c
From the textbook solve the problems 17, 18, 19, 22, 23 and 24 from the Chapter 4.
Solve the problems 21, 22, 24, 30 from the Chapter 4 additional exercises at
http:/www.athenasc.c
Mathematics 170B HW1 Due Thursday, January 6, 2011. Problems 1, 2, 3, 4 on page 246. (Note: On problem 4, the PDF of X is general not the uniform from Example 3.14 in Chapter 3.) The following problems are from my 170A nal exam last Fall. They are the one
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3
c
c
From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4.
Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at
http:/www.athenasc.com
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6
c
c
Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the problems below:
Problem 1. Denote points P0 ,
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4
c
c
solutions
From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4.
Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7
addi
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6
c
c
Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at
http:/www.athenasc.com/prob-supp.html
And also the problems below:
Problem 1. Denote points P0 ,
Midterm 1 practice, Math 170b - Lec 1, Winter 2013
Instructor: Toni Antunovi
c
c
Name and student ID:
Question
Points
1
10
2
10
3
10
4
10
5
10
Total:
50
Score
1. (a) (2 points) If is a sample space, what is P().
Solution: P() = 1
(b) (2 points) If P(A) =
T. Liggett
Mathematics 170B Midterm 2 Solutions
May 23, 2012
(20) 1. (a) State Markovs inequality.
Solution: If X 0, then P (X a) EX/a for a > 0.
(b) Prove Markovs inequality.
Solution: a1cfw_X a X . Taking expected values gives aP (X a) EX .
(c) Suppose
Mathematics 170B Selected HW Solutions.
F4 . Suppose Xn is B (n, p).
(a) Find the moment generating function Mn (s) of (Xn np)/
np(1 p).
Write q = 1 p. The MGF of Xn is (pes + q )n , since Xn can be
written as the sum of n independent Bernoullis with para
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 2
c
c
From the textbook solve the problems 17, 18, 19, 22, 23 and 24 from the Chapter 4.
Solve the problems 21, 22, 24, 30 from the Chapter 4 additional exercises at
http:/www.athenasc.c
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 5
c
c
From the textbook solve the problems 4 and 5 from the Chapter 5.
Solve the problems 2, 4, 6, 7, 9 and 10 from the Chapter 7 additional exercises at
http:/www.athenasc.com/prob-supp
170B Homework 3- Due Friday, October 16th.
Homework should be written neatly and clearly explained. If it requires more than one
sheet, the sheets must be stapled. Include your name and id number in the top right corner
of your homework.
From the textbook
Math 170B Probability Theory: Lecture 3
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
January 8th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
January 8th 2016
1/9
Sums of Independent Random Variables
The last example we talke
Math 170B Probability Theory: Lecture 13
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
February 12th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
February 12th 2016
1/6
The Central Limit Theorem
In the weak law of large number
Math 170B Probability Theory: Lecture 4
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
January 11th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
January 11th 2016
1/9
Covariance
In Chapter 2 we learned the definition of varianc
Math 170B Probability Theory: Lecture 4
Yuan Zhang
yuanzhang@math.ucla.edu
Department of Mathematics
UCLA
January 11th 2016
Yuan Zhang (Dept. Math, UCLA)
Probability Theory
January 11th 2016
1/9
Covariance
In Chapter 2 we learned the definition of varianc
Mathematics 170B - Homework 4
April 18, 2016
Problems 29, 30, 32, 33, on pages 256-257.
I. Suppose that X is a Gamma random variables, with parameters and
= 1. That means X has the following probability density function
fX (x) = 2 xex , x > 0.
Find its m
Mathematics 170B - Homework 2
April 4, 2016
Problems 8,9,10,12 on pages 246-247.
I. Let X1 , X2 , , Xn be i.i.d. random variables with mean and variance
2 , and let
X 1 + X2 + + Xn
X=
n
be the sample mean. Show that for any i, Xi X and X are uncorrelated