MathUA.233.001: Theory of Probability
Homework 2
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
Theory of Probability : Recitation 2(Feb13)
1. Solutions for examples
How many 5digit numbers can be formed from the integers 1, 2,., 9 if no digit can appear
more than twice? (For instance, 41434 is not allowed)
1.
Solution. There are only three types o
MathUA.233.001: Theory of Probability
Homework 4
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
MathUA.233.001: Theory of Probability
Homework 3
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
Theory of Probability : Recitation 6 (Mar 13)
1. Solutions for selected problems from Assignment 6
1.
We shall suggest two dierent solutions.
Solution 1.
Let
0, 1, 2, , n
X
be a random variable given by
X = B.
Then, possible values of
X
are
and by Bayes
MathUA.233.001: Theory of Probability
Homework 7
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
MathUA.233.001: Theory of Probability
Homework 6
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
MathUA.233.001: Theory of Probability
Homework 5
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
Theory of Probability : Recitation 4 (Feb27)
1. Solutions for missing problems
Example 2.
head,
E2
A
pbiased
coin is tossed
6
times. Let
E1
be the event 5 or more outcomes are
the event the rst two outcomes are head and
E3
the event the last two outcomes
MathUA.233.001: Theory of Probability
Final cheatsheet
The nal will be a closedbook exam, so there are some facts from the course
that youll be expected to remember. It will not be assumed that you remember
everything from all the classes. Try to make s
MathUA.233: Theory of Probability
Lecture 5
Tim Austin
[email protected]
cims.nyu.edu/tim
Conditional probability (Ross Ch 3)
Sometimes we obtain partial information about an experiment,
and want to compute probabilities in the light of that information.
MathUA.233: Theory of Probability
Lecture 6
Tim Austin
[email protected]
cims.nyu.edu/tim
1 / 22
From last time. 1
If F S is an event with P(F ) > 0, and E S is another event,
then
P(E F )
.
P(E  F ) =
P(F )

cfw_z
definition
STORY: As the experiment i
MathUA.233: Theory of Probability
Lecture 2
Tim Austin
[email protected]
cims.nyu.edu/tim
1 / 28
First words
Almost every aspect of life contains an element of chance.
In probability theory we learn to reason quantitatively about
chance. Nowadays, probabi
MathUA.233: Theory of Probability
Lecture 4
Tim Austin
[email protected]
cims.nyu.edu/tim
Reminder: the axioms again
Axiom 1: Any event E satisfies 0 P(E) 1.
Axiom 2: P(S) = 1.
Axiom 3: If the sequence of events E1 , E2 , . . . are disjoint
(= mutually ex
MathUA.233: Theory of Probability
Lecture 3
Tim Austin
[email protected]
cims.nyu.edu/tim
From last time. 1
To model an experiment, we choose:
I
A sample space S, containing all possible outcomes
(according to some description).
I
A probability value P(E)
MathUA.233: Theory of Probability
Basic facts from combinatorial analysis
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
This handout summarizes the facts from combinatorial analysis that we will
need in Theory of Probability.
I will
MathUA.233.001: Theory of Probability
Supplement: dening independence for
several events
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
In all of the following, is the sample space of some experiment and P is a
probability distributi
MathUA.233.001: Theory of Probability
Homework 1
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
1. (3 marks) Let E be an event and F1 , F2 , . . . a sequence of events (that is,
these are all subsets of some sample space). Prove rigo
Theory of Probability : Recitation 10(Apr 17)
Throughout this article, for any possible random variable X , fX (x) and FX (x) denote the PDF
and CDF of X , respectively. We rst provide two basic and useful Lemmas. The rst lemma is a
formula for the unifor
X (, 2 ) Y = X i.e.,
(0, 12 ) = 70 = 10 Y = X70
10
P (X > 50) = P
X 70
> 2
10
= P (Y > 2) = 1 (2) = 1 0.0228
0.977.
P (60 < X < 80) = P
1 <
X 70
<1
10
= P (1 < Y < 1) = (1) (1)
= 0.8413 0.1587
f (x, y) > 0
y
f (x,
2
y
c(y
MathUA.233.001: Theory of Probability
Homework 9
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
Theory of Probability : Recitation 9(Apr 10)
1. Compute probability density functions
Example 1.
variables.
(a)
Let U be a uniform random variable on (0, 1). Find the PDFs of the following random
U
(b) log(1 U )
Example 2.
Let X be a continuous random var
MathUA.233.001: Theory of Probability
Homework 8
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or po
MathUA.233.001: Theory of Probability
Homework 10
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or p
Theory of Probability : Recitation 12(May 1)
1. Reviews
(1) Change of Variables
Suppose that we know the joint PDF fX, Y (x, y) of X, Y . If two RVs U and V are obtained from
X and Y in a way that
U = F (X, Y ),
(0.1)
V = G(X, Y )
for some function F and
MathUA.233.001: Theory of Probability
Midterm cheatsheet
The midterm will be a closedbook exam, so there are some facts from the
course that youll be expected to remember. It will not be assumed that you remember everything from all the classes. Try to
MathUA.233.001: Theory of Probability
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Lectures: Tuesdays and Thursdays 9.30 10.45, in Warren Weaver 101.
Recitations: There are two recitation sections: 233.002 and 233.003. The rst is
l
Theory of Probability : Recitation 13(May 08)
1. Reviews
(1) Markov's Inequality
When a random variable Y has somewhat complicate structure so that we can not compute the
proabability P (Y > a), we can estimate it via the following wellknown Markov's ine
MathUA.233.001: Theory of Probability
Homework 12
Tim Austin
803 Warren Weaver Hall
[email protected]
cims.nyu.edu/tim
Questions or parts of questions which are marked with an asterisk are probably
more challenging, and are purely for your education, or p