AMS 507: Introduction to Probability
Instructor: Jiaqiao Hu Class: Tuesdays and Thursdays 11:20am 12:40pm, Harriman Hall 137 Office: Math Tower 1107 Phone: (631)6328239 Email Address: [email protected] Office Hours: Wednesdays 12:30pm 2:30pm or by ap
Common Quals Probability Questions for OR and Stat Tracks, Fall 2004
DO THREE OUT OF FOUR QUESTIONS (If you do all four, only the first three problems will be graded) 1. A stick of length 1 is cut into two pieces at a randomly chosen point uniformly dist
AMS507 Introduction to Probability. First Midterm.
October 13, 2009 Your name: Your ID number: You shall receive 10 points for each of the following 10 question. The total will be divided by 10. The maximum score is 10. 1. How many hands of 13 cards dealt
390 Chapter 8 Limit Theorems
Remark W hen all t he Pi a re e qual t op, X is a b inomial r andom v ariable. Hence,
t he p receding i nequality shows t hat, f or a ny s et o f n onnegative i ntegers A ,
I
Summary
Two useful probability bounds are provided
A First Course in Probability 271
Problems
are inde
lorn varin can b e
nal ranI a i2, 1. =
I
: f.Li a nd
'1
6.1. Two fair dice are rolled. Find the joint probability mass
function o f X a nd Y w hen
(a) X is t he largest value obtained o n any die and Y
352 Chapter 7 Properties of Expectation
random variable and, second, t hat t he m oment g enerat~X;
ing function o f t he sum o f i ndependent random variables
X=Ln
is e qual t o t he product o f t heir m oment g enerating funci =l
tions. These results le
A First Course in Probability 163
An i mportant p roperty o f t he expected value is t hat t he
expected value o f a s um o f r andom variables is e qual t o
the sum o f t heir expected values. T hat is,
n
= L E[X;]
i =l
Problems
4.1. Two balls are chosen
A First Course in Probability 97
a nd y ielding t he r esult
pH E E
( tl
1
_
z) 
P (EziHt)P(HtiEl)
2:7= 1 P (EziHt)P(HtiEl)
F or i nstance, s uppose t hat o ne o f t wo c oins is c hosen t o b e flipped. L et H t b e t he
e vent t hat c oin i, i = 1, 2,
AMS 507
Lecture 5
Chapter Three
Conditional Probability and Independence
3.2 Conditional Probabilities
Definition: If P ( F ) > 0, the conditional probability of E given F, denoted by P(EF), is
P ( EF )
P( E  F ) =
.
P( F )
This definition satisfies the
AMS 507
Lecture 2
Fall 2013
Generalized Basic Principle of Counting:
If r that are to be performed are such that the first one may result in n1 possible outcomes,
and if for each of these n1 possible outcomes there are n2 possible outcomes of the second
e
Your name:
Your ID number:
AMS507 Introduction to Probability  Final Exam
Fall 2009
You shall receive 5 points for each of the following 20 questions.
1. There are 5 red balls, 4 green balls, and 3 white balls. In how many dierent ways can these
balls be
AMS 507, CET 551, Lecture 3
Fall Semester, 2013
Example: Consider a set of n antennas of which m are defective and nm are functional and assume that all
of the defectives and all of the functionals are considered indistinguishable. How many linear orderi
AMS 507, Lecture 6 Handout
3.4. Independent Events
Definition: Two events E and F are called independent if
P ( EF ) = P ( E ) P ( F ).
If two events are not independent, they are called dependent. If E and F are independent,
we say that cfw_E, F is an in
Your name:
Your ID number:
AMS507 Introduction to Probability. Final Test
December 18, 2008
You shall receive 5 points for each of the following 20 questions.
1. In how many ways can a man divide 7 gifts among his 3 children if
the eldest is to receive 3
AMS 507
Introduction to Probability
Fall Semester
Chapter One
Combinatorial Analysis
I will go over this chapter lightly because it covers material from combinatorics, and
many have taken classes in this area. For those of you taking the actuarial examina
Your name:
Your ID number:
AMS507 Introduction to Probability. Final Test
December 17, 2007
You shall receive 5 points for each of the following 20 questions.
1. Suppose that a standard deck of 52 cards is shued and the cards
are then turned over one at a
A Solution Manual for: A First Course In Probability by Sheldon M. Ross.
John L. Weatherwax December 16, 2011
Introduction
Acknowledgements
Special thanks to (most recent comments are listed first): John Williams (several contributions to chapter 4), Timo
A First Course in Probability IS
Summary
The basic principle o f c ounting states t hat if an experiment
consisting o f t wo phases is such t hat t here a re n possible
outcomes o f p hase 1 and, for each o f t hese n o utcomes,
there are m possible outco
48 Chapter 2 Axioms of Probability
I f S is finite and each one point set is assumed to have
equal probability, then
P (A)
=~
where lEI d enotes the number o f outcomes in the event E.
P (A) can b e i nterpreted either as a longrun relative
frequency o r
2 12 C hapter 5 Continuous Random Variables
Problems
5 .1. L et X b e a r andom v ariable with probability density
function
1 < X < 1
o therwise
f (x)  cfw_c(1  x2)

0
(a) W hat is t he v alue o f c?
( b) W hat is t he c umulative distribution functio
430 Chapter 10 Simulation
Summary
L et F b e a continuous distribution function a nd U a uniform (0, 1) random variable. Then the random variable
p  1 (U) has distribution function F, w here F  1 (u) is that
value x such that F(x) = u. Applying this res
4 12
Chapter 9 Additional Topics in Probability
Let X b e a random variable that takes o n o ne o f
n possible values according to the set o f probabilities
cfw_pJ, . . , pnl The quantity
n
H (X) = 
I)i log2 (pi)
is called the entropy of X . I t c an b e
J. Appl. Prob. 46, 12091212 (2009)
Printed in England
Applied Probability Trust 2009
AN INEQUALITY FOR VARIANCES
OF THE DISCOUNTED REWARDS
EUGENE A. FEINBERG and
JUN FEI, Stony Brook University
Abstract
We consider the following two denitions of discount
AMS 507, Lecture 7
Chapter Four: Random Variables
4.1. Random Variables
Definition: Let S be the sample space of an experiment. A realvalued function
X : S R is called a random variable of the experiment if, for each interval
I R , cfw_ s : X ( s ) I is
AMS 507
Lecture 4
2.6. Probability as a Continuous Set Function
A sequence of events cfw_E n , n 1 is said to be an increasing sequence if
E1 E 2 E n E n +1 . If cfw_E n , n 1 is an increasing sequence of events, then
we define lim E n = E i .
n
i =1
A s
Chapter Eight: Limit Theorems
Some Basic Facts
Proposition 2.1 Markovs inequality
If X is a random variable that takes only nonnegative values, then for any value a>0,
E[ X ]
Pcfw_ X a
.
a
Proposition 2.2 Chebyshevs inequality
If X is a random variable w
Continuous Random Variables
5.4 Normal Random Variables
Denition of Normal Random Variables
We say the X is a normal random variable, or simply that X is normally
distributed, with parameters and 2 if the density of X is given by
2
2
1
e(x) /2
f (x) =
2
Math 461, Solution to Written Homework 10
1. (4 points) The moment generating function of X is given by MX (t) = exp(2et 2)
and that of Y by MY (t) = ( 43 et + 14 )10 . If X and Y are independent, find
(a) P(X + Y = 2);
(b) P(XY = 0);
(c) E[XY ].
Solution