6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
November 08, 2010
Problem Set 10
Problem 1. [15 points] Suppose Pr cfw_ : S [0, 1] is a probability function on a sample space, S, and let B be an event such that Pr cfw_B > 0
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 14, 2008
Problem Set 7
Problem 1. [15 points] Express
n i=0
i2 x i
as a closed-form function of n. Problem 2. [20 points] (a) [5 pts] What is the product of the first
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11
11.1
Cardinality Rules
Counting One Thing by Counting Another
How do you count the number of people in a crowded room? You could count heads, since for each person there is exactly one head. Alternatively,
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 14, 2010
Problem Set 2
Problem 1. [12 points] Define a 3-chain to be a (not necessarily contiguous) subsequence of three integers, which is either monotonically incr
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 27, 2010
Midterm
Name:
This quiz is closed book, but you may have one 8.5 11" sheet with notes in your own handwriting on both sides. Calculators are not allowed. You
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14
14.1
Events and Probability Spaces
Let's Make a Deal
In the September 9, 1990 issue of Parade magazine, columnist Marilyn vos Savant responded to this letter: Suppose you're on a game show, and you're given
"mcs-ftl" - 2010/9/8 - 0:40 - page 417 - #423
15
15.1
Conditional Probability
Definition
Suppose that we pick a random person in the world. Everyone has an equal chance of being selected. Let A be the event that the person is an MIT student, and let B be
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16
16.1
Independence
Definitions
Suppose that we flip two fair coins simultaneously on opposite sides of a room. Intuitively, the way one coin lands does not affect the way the other coin lands. The mathematic
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17
Random Variables and Distributions
Thus far, we have focused on probabilities of events. For example, we computed the probability that you win the Monty Hall game, or that you have a rare medical condition
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18
18.1
Expectation
Definitions and Examples
The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the a
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19
Deviations
In some cases, a random variable is likely to be very close to its expected value. For example, if we flip 100 fair, mutually-independent coins, it is very likely that we will get about 50 heads.
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20
Random Walks
Random Walks are used to model situations in which an object moves in a sequence of steps in randomly chosen directions. Many phenomena can be modeled as a random walk and we will see several e
"mcs-ftl" - 2010/9/8 - 0:40 - page 379 - #385
13
Infinite Sets
So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite set does have an infinite number of ele
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12
Generating Functions
Generating Functions are one of the most surprising and useful inventions in Dis crete Math. Roughly speaking, generating functions transform problems about se quences into problems abo
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10
Recurrences
A recurrence describes a sequence of numbers. Early terms are specified explic itly and later terms are expressed as a function of their predecessors. As a trivial example, this recurrence descr
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1
Propositions
Definition. A proposition is a statement that is either true or false. For example, both of the following statements are propositions. The first is true and the second is false. Proposition 1.0.1.
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2
2.1
Patterns of Proof
The Axiomatic Method
The standard procedure for establishing truth in mathematics was invented by Eu clid, a mathematician working in Alexandria, Egypt around 300 BC. His idea was to begi
"mcs-ftl" - 2010/9/8 - 0:40 - page 43 - #49
3
Induction
Now that you understand the basics of how to prove that a proposition is true, it is time to equip you with the most powerful methods we have for establishing truth: the Well Ordering Principle, the
"mcs-ftl" - 2010/9/8 - 0:40 - page 81 - #87
4
Number Theory
Number theory is the study of the integers. Why anyone would want to study the integers is not immediately obvious. First of all, what's to know? There's 0, there's 1, 2, 3, and so on, and, oh ye
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5
Graph Theory
Informally, a graph is a bunch of dots and lines where the lines connect some pairs of dots. An example is shown in Figure 5.1. The dots are called nodes (or vertices) and the lines are called e
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6
6.1
Directed Graphs
Definitions
So far, we have been working with graphs with undirected edges. A directed edge is an edge where the endpoints are distinguished-one is the head and one is the tail. In partic
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7
Relations and Partial Orders
A relation is a mathematical tool for describing associations between elements of sets. Relations are widely used in computer science, especially in databases and scheduling appl
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9
Sums and Asymptotics
Sums and products arise regularly in the analysis of algorithms, financial applica tions, physical problems, and probabilistic systems. For example, we have already encountered the sum 1
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
December 8, 2010
Notes for Recitation 23
Theorem 1. Let E1 , . . . , En be events, and let X be the number of these events that occur. Then: Ex (X) = Pr cfw_E1 + Pr cfw_E2 +
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 10, 2010
Problems for Recitation 1 1 Team Problem: A Mystery
A certain cabal within the 6.042 course staff is plotting to make the final exam ridiculously hard. ("Pr
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 1, 2010
Problems for Recitation 7 1 A Protocol for College Admission
Next, we are going to talk about a generalization of the stable marriage problem. Recall that we h
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 1, 2010
Notes for Recitation 7 1 A Protocol for College Admission
Next, we are going to talk about a generalization of the stable marriage problem. Recall that we have
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 6, 2010
Problems for Recitation 8 1 Build-up error
Recall a graph is connected iff there is a path between every pair of its vertices. False Claim. If every vertex in
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 5, 2010
Notes for Recitation 8 1 Build-up error
Recall a graph is connected iff there is a path between every pair of its vertices. False Claim. If every vertex in a g
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 8, 2010
Problems for Recitation 9 1 Traveling Salesperson Problem
Now we're going to talk about a famous optimization problem known as the Traveling Sales person Probl