Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
Math 411 Problem Set 1
Issued: 09.02
Due: 09.09
1.1. Two dice are rolled. What is the probability that (a) the two numbers
will dier by 1 or less and (b) the maximum of the tw
Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
Math 411 Problem Set 2
Issued: 09.09
Due: 09.16
2.1. A bet is said to carry 3 to 1 odds if you win $3 for each $1 you bet.
What must the probability of winning be for this to
1.2. We have P (A B) = P (A) + P (B) P (A B). Since P (A B) 1, this implies that
P (A B) P (A) + P (B) 1 = 0.1. We also have P (A B) min(P (A), P (B) = 0.4. It is easy
to see that 0.1 and 0.4 are the minimal and the maximal possible value of P (A B). The
Mathematical Probability, Math 411 - Homework 6
From the textbook solve the problems 1 and 2 at the end of the Chapter 3.
Solution to Problem 1: The PMF of Y is
P(Y = 1) = P(X 1/3) = 1/3, P(Y = 2) = P(X > 1/3) = 2/3,
so E(Y ) = 1 1/3 + 2 2/3 = 5/3.
Using
Mathematical Probability, Math 411 - Homework 2 solutions
Problem 1. A person places randomly n letters in to n envelops . What is the probability that
exactly k letters reach their destination.
Solution See online notes in the course website.
Problem 2.
Mathematicl Probability, Math 411, Spring 2016 - Homework 7 solutions
From the textbook
Solution to Problem 6: Let X be her waiting time then for t > 0
FX (t) = P(X t) = P(X t|no customer in front)P(no customer in front)
+ P(X t|no customer in front)P(no
Probability Theory, Math 411 - Homework 3 solution
Problem 1. The dorm in which you live houses 1% of the total TAMU
student population. You know 30% of the students living in your dorm, but
you know only 2% of the rest of TAMU student population. A lot o
Analysis exercise 4
Sequences of functions
February 5, 2014
Due Wednesday Feb 12.
1. Do problems 9.1 9.9, from the textbook.
The a b symbols means all the questions in the closed interval [a, b]
2. Let f n : (a, b) R be a sequence of continuously differen
Midterm 2 practice, Math 411
1. (10 points) We have a biased coin (probability of heads equal to p). In the first stage
of the experiment we keep tossing it until we get a heads, and we remember how many
times we had to toss it (say k times). In the 2nd s
Lecture Notes in Probability
Eviatar B. Procaccia
Department of Mathematics
Texas A&M University
Based on lecture notes written by Professor Raz Kupferman,
The Hebrew University of Jerusalem.
April 15, 2016
2
Contents
1
2
3
Basic Concepts
1
1.1
The Sample
Mathematical Probability, math 411 homework 8 - solutions
Problem 1. Let X be exponentially distributed with parameter . Find
the PMF of Y = dXe, where dxe for a real number x is the rounding of
x to the nearest integer whose value is greater or equal to
SYLLABUS
"
Mathematical Probability - Math 411
Spring 2016
MWF 9:10-10:00pm, Blocker 149
https:/sites.google.com/site/ebprocaccia/teaching/2016-spring-411
Course Description and Prerequisites
Probability theory is used in many fields, e.g. economics, comp
Mathematical Probability Math 411 - Homework 9
Problem 1. Show that for random variables X, Y and Z we have
E[E[E[X|Y ]|Z] = E[X].
Apply this formula to the following problem: Roll a far 6-sided die and observe the number Z
that came up. Then toss a fair
Midterm 1 practice
1. All your HW exercises.
2. (10 points) Suppose a random variable has the following PMF:
PX (k) =
c
,
k2
k cfw_1, 2, 3.
Find c, E[X].
3. (a) (6 points) If = A B, P(A B c ) = 0.6, P(Ac B) = 0.2 find the probabilities of A and B.
(b) (6
Mathematical Probability, Math 411, Spring 2016 - Homework 11 (do not hand in)
Problem 1. It is known that among TAMU students 60% support candidate A for the student
council, while only 40% support candidate B. In order not to waste the time of too many
Mathematical Probability, Math 411, Homework 8
Remember the convolution formula
for the sum of independent random variR
ables Z = X + Y : fZ (z) = fX (x)fY (z x)dx.
From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11,14 at the end of
the Chapter 4.
Mathematical Probability, Math 411, Spring 2016 - Homework 10
Problem 1. Show that Almost sure convergence does not imply mean square convergence.
Hint: Let Xn be independent with
1
x = n3
n2
pXn =
.
x=0
1 n12
Let X = 0. Calculate the mean square differe
Mathematical Probability, Math 411 - Homework 9
From the textbook solve the problems 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.com/prob-supp.html
And also the problems
Analysis exercise 3
Continuous functions
January 22, 2014
Due Wednesday Jan 29.
1. Do problems 4.50 4.53, from the textbook.
The a b symbols means all the questions in the closed interval [a, b]
2. Let F X be closed subset in a metric space.
Prove the fun
Homework 7 (Midterm 2), Math 131B - Winter 2014
Instructor: Eviatar B. Procaccia
P
n
1. (a) Define the radius of convergence of a power series
n=0 an x
(b) What is the interval of convergence for the following power series:
n
P
1. n=1 (x+1)
.
n
2.
3.
P
1
1. The number of heads is even if it is 0, 2, or 4. It follows that the probability is 32
(1 + 52 +
1+10+5
1
= 16
4 )=
32
32 = 2 .
Another solution: it is clear that the probability is the same as probability that the number of
tails is even. But if th
MATH 411 Sections 200 & 502 - Fall 2016
Homework Assignment 3
Due: Tuesday September 27. In class.
Section and problem numbers are from the course textbook:
D. Bertsekas & J. Tsitsiklis, Introduction to Probability, Second Edition.
Hand In:
1. Problems 14
MATH 411 Sections 200 & 502 - Fall 2016
Homework Assignment 2
Due: Tuesday September 20. In class.
Section and problem numbers are from the course textbook:
D. Bertsekas & J. Tsitsiklis, Introduction to Probability, Second Edition.
Hand In:
1. Problems 13
Math 411 Syllabus
Course title and number
Term
Class time and location
Math 411 - Mathematical Probability
Fall 2016
Sections 200 & 502: TR 8:00-9:15am BLOC 163
INSTRUCTOR INFORMATION
Name
Michael Brannan, Assistant Professor
Phone Number
Department of Ma
MATH 411 Sections 200 & 502 - Fall 2016
Homework Assignment 1
Due: Thursday September 8. In class.
Section and problem numbers are from the course textbook:
D. Bertsekas & J. Tsitsiklis, Introduction to Probability, Second Edition.
Hand In:
1. Problems 2
Mathematical Probability, Math 411 - Homework 5 spltions
From the textbook solve the problems 16, 22, 24 at the end of the Chapter 2.
Solution to Problem 16:
(a) To find a use the condition that the probabilities must add up to 1:
pX (3) + pX (2) + pX (1)
Analysis exercise 5
Complete metric spaces
February 13, 2014
Due Wednesday Feb 19.
1. Do problems 4.66 4.72, from the textbook.
The a b symbols means all the questions in the closed interval [a, b]
2. Write a fullproofof the Arzela-Ascoli
theorem:A closed
Analysis List of Questions
March 14, 2014
1. Problems 3.26, 3.283.33, 3.353.42, 4.294.32, 4.504.53, 9.19.9, 4.664.72, 9.309.32,
9.36 9.37 from the textbook.
The a b symbols means all the questions in the closed interval [a, b].
1 |xi yi |
2. Prove that i