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
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, 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 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
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
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
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, 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:
Practice Midterm 2 version 2, Math 170b, 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
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
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 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) =
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 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
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 ,
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
Practice Midterm 2 version 2, Math 170b, 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
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 ,
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 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
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
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 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
INTRODUCTION
TO
PROBABILITY
by
Dimitri P. Bertsekas and John N. Tsitsiklis
CHAPTER 1:
ADDITIONAL PROBLEMS
Last updated: September 12, 2005
SECTION 1.1. Sets.
Problem 1. We are given that P (A) = 0.55, P (B c ) = 0.35, and P (A B) = 0.75.
Determine P (B) a
170B Probability Theory
Puck Rombach
Solutions
Problem 1
A nonnegative integer-valued random variable X has one of the following two expressions as its transform:
1. M(s) = e2(e
e s 1 1)
,
2. M(s) = e2(e
e s 1)
.
(a) Explain why one of the two cannot poss
Mathematics 170B - Homework 1 : Due
Thursday, Jan. 19, 2017.
Problems 1, 2, 3, 7 on Pages 186-192.
A. A coin with probabiltiy p of heads is tossed till the first head occurs. It
is then tossed again till the first tail occurs. Let X be the total number of
170B Killip
Homework 2
Due Wed, April 19
Recall (from class) the definition of the 2 random variable: X 2 if and only if
(
1
x 2 1 ex/2 : x 0
2/2 (/2)
fX (x) =
0
: otherwise
R 1 x
where > 0 and () = 0 x e dx for all > 0.
(1) Suppose X 2 and Y 2k are indep
170B Killip
Homework 1
Due Wed, April 12
(1) Suppose X Exponential(). Find the PDF of Y = sin2 (X).
Note: While the solution involves an infinite sum, it is an example of a geometric
series and so can be evaluated.
(2) Let X1 and X2 be independent and uni
10. Strong Convergence
170B Probability Theory, Puck Rombach
Last updated: November 23, 2016
Bertsekas & Tsitsiklis: Section 5.5. Assumed knowledge: Convergence in probability, convergence in distribution.
10.0.1
Almost Sure Convergence
Let cfw_Xn be a s
5. Sums of a random number of random variables
170B Probability Theory, Puck Rombach
Last updated: November 20, 2016
Bertsekas & Tsitsiklis: Section 4.5. Assumed knowledge: MGFs and conditional expectation/variance.
Consider the sum X = X1 + X2 + . . . +
170B Probability Theory
Homework Sheet 2
Puck Rombach
due: 10/7/16
Problem 1
Let X be an exponential random variable with fX (x) = ex , and let Y be a uniform continuous
random variable on the interval [2, 3]. Find the PDF of the independent sum Z = X + Y