CS206
Apr. 15, 2014
HW4 - Some Solutions
Question 2: I will just do X and Z. Lets dene a matrix for X using X(i, j) = i + j, where
i, j = 1, . . . , 6. The entries in ROW i give the sum of the faces when the rst die shows i and the
entries in COL. j show
CS206
HW1
Sept. 8, 2009
(Write up convincing answers to the questions marked by an asterisk (*). Some of the unstarred questions will be worked in recitation. (*) indicates a more challenging problem, not required to hand in unless you want feedback - its
CS206
Review Sheet 5
April 26, 2014
1. The Random Walk on the Integers: We start with the coin toss game which uses a
coin with probability P for Head and probability q = 1 P for Tail. You get a dollar if it
comes up Head and lose a dollar it it shows Tai
CS206 - Discrete Structures II
Midterm Exam - Section 01 & 02 - Fall 2015
Name:
Student I.D.:
1
Answer Key for Exam A
2
Problems
This test is CLOSED book, CLOSED notes, NO calculators, NO cheat sheets.
You have 1:20 hours to answer the questions.
Answe
CS206
Review Sheet 4
April 13, 2014
1. Variance: Suppose X is a random variable on a probability space (S, P ) and with expected value E(X) = m. The variance of X is the expected squared deviation from m,
dened by
V (X) = E([X m]2 ).
(1)
Evaluating (1) ov
Independence
Reading: Ross, Ch 3., Sec. 4.
02/25/2009
Monday, March 9, 2009
CS206 - Intro. to Discrete Structures II
1
Example 1
From a deck of 52 cards we draw one card randomly. What is the probability that this card will be an ace? What is the probabi
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 2 Solutions
1. From a group of 3 freshmen, 4 sophomores, 4 juniors, and 3 seniors a committee of size 4 is
randomly selected. We assume that all people are distinguisha
CS 206 Midterm I, Spring 2015
Prof. David Cash
Write your answers in the space provided in each problem. An extra blank sheet is
included after the problems. If you want the extra sheet or a back of a test sheet
graded, indicate this very clearly in the s
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Recitation 6 Solutions
1. You are going to play 2 games of chess against one opponent of unknown strength randomly
drawn from a club where 10% of players are masters, 30% are in
CS206
OLD TEST
SOME SOLS.
1a: 2(k!)(k!)/(2k)!. There are (2k)! ways to order the n = 2k integers. If the odds and evens
alternate, there are k! ways to order EACH of these subsets and 2 ways to alternate them,
ONE if an odd integer is rst and the OTHER i
CS206
March 12, 2014
TEST 1
Instructions:
You may use ONE page of prepared notes (both sides), but otherwise the test is closed
book. All work must be your own.
Show ALL your work. You will get little or no credit for an unexplained answer.
Do all your
CS206
HW4 (* due date is November 4, 2009)
October 20, 2009
1. A fair die is tossed twice. Let X = the sum of the faces, Y = the maximum of the two faces, and Z = |face 1 - face 2|. (a) Write down the value of X,Y , Z, and W = XZ for each outcome w S. (b)
CS206
October 27, 2010
TEST 1
Instructions:
You may use ONE page of prepared notes (both sides), but otherwise the test is closed
book. All work must be your own.
Show ALL your work. You will get little or no credit for an unexplained answer.
Do all yo
CS206
HW3
March 13, 2014
Hand-in problems with (*) by March 31, 2014. (+) ones are interesting. (*) is more challenging.)
1. Eight people are seated at random in a row of eight seats.
(a) Describe the sample space S and nd |S|, its size.
(b) Find the prob
CS206
December 19, 2008
FINAL TEST
Instructions:
Do all your work in the blue exam books.
You may use two pages of prepared formulas and notes, but all work must be
your own.
Please write your answers IN THE GIVEN ORDER, though you may solve problems i
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Homework 5 Solutions
1. Let X be uniformly distributed on the set C = cfw_2, 1, 0, 1, 2. How would you write X as a
a fuction of a Uniform Random Variable? Draw the graph of X.
Rutgers University
CS206: Introduction to Discrete Structures II, Summer 2016
Recitation 7 Solutions
1. Suppose that there are two types of drivers: good drivers and bad drivers. Let G be the event
that a certain person is a good driver, A be the event th
1. February 20th, 2016 - Lecture:
Independence
1.
Topics
Multiplication rule for conditional probability.
Independent events: two equivalent definitions and several examples with cards and dice.
People v. Collins and the prosecutor's fallacy.
Mutual indep
Homework # 2
1. (8 points) Prove that P(AB)P(A)+P(B)P(AB)P(A)+P(B) for any events A
and B. Prove the general version by induction, which says that
if A1,.,AnA1,.,An are events
then P(ni=1Ai)P(ni=1Ai)P(i=1nAi)P(i=1nAi). When does this
o
inequality become a
- Homework 7 answers
1.
(1.5 points each) Find the generating function for the sequence a0,a1,a2,
where ak is each of the following. Your solution does not need to be closed form.
1.
ak= the number of solutions to e1+e2+e3=k, where 0ei4 for each i.
(1+x1+
1. March 13th, 2016 - Lecture:
Linearity of expectation
1.
Introduction
Topics:
o
Linearity of expectation.
o
Expectation of binomial, geometric, and negative binomial frequency functions.
Applications to computing expectations:
o
Birthday problem,
o
ball
Here are some overall notes for Discrete II
Starting with Distributions:
A random variable is a variable that is able to dynamically measure the number of times an
event occurs during a particular experiment.
- So if I were to flip a coin 5 times, I can m
1.
2.2 Sample Space and Events
The set of all possible outcomes of an experiment is known as the sample
space of the experiment and is denoted by S. Examples:
1.
The sex of a newborn child: S = cfw_g, b
2.
Horse race with seven horses: S = cfw_all 7! perm
1. April 15th, 2016 - Lecture
o
Idea: To solve a1,a2,.a1,a2,.
Part 1: Define generating function
A(x)=k=0A(x)=k=0
"Pull out" the initial conditions
Substitute recurance relation formula
Expand, algebra, etc until you can substitute
Get
A(x)A(x)
A(x)=A(x)=
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2017
Professor David Cash
Homework 8
Due at 9am on Friday, Mar 31
Instructions:
Follow the collaboration policy stated in class: You may discuss problems together but you
must write
Rutgers University
CS206: Introduction to Discrete Structures II, Spring 2017
Professor David Cash
Homework 7
Due at 9am on Tuesday, Mar 28
Instructions:
Follow the collaboration policy stated in class: You may discuss problems together but you
must writ
CS 206: Probability
Instructor: Pranjal Awasthi
Announcement
HW 3 out today. Due March 22nd.
Last Lecture
=
()
Last Lecture
A and B are independent events if = ()
Last Lecture
1 , 2 , 3 , are independent events if for all =
2,3, , and for all indice
CS 206: Probability
Instructor: Pranjal Awasthi
Textbooks
Probability with Applications in Engineering, Science, and
Technology
M. A. Carlton and J. L. Devore
Available for free through university library website.
Other Textbooks
S. Ross, A First Cour
CS 206: Probability
Instructor: Pranjal Awasthi
Announcement
HW 3 due tomorrow.
Last Lecture
Law of Total Probability
Bayes Theorem
Law of Total Probability
If 1 , 2 , are disjoint events such that = ,
then for any event
= 1 1 + 2 (2 ) + + ( )
Monty
CS 206: Probability
Instructor: Pranjal Awasthi
Announcement
Quiz next Friday.
Expectation
A fair coin is tossed 100 times. What is the average
number of heads seen?
Expectation
A fair coin is tossed 100 times. What is the average
number of heads seen?
CS 206: Probability
Instructor: Pranjal Awasthi
Announcement
Quiz on Friday.
Random Variables
Given experiment and sample space , a random
variable associates a number to every outcome in .
Random Variables
Two types: discrete and continuous
Random Var