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
"mcs-ftl" - 2010/9/8 - 0:40 - page 313 - #319
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 13, 2010
Notes for Recitation 10 Analysis of Two Networks
Two communication networks are shown below. Complete the table of properties and be prepared to justify your
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 13, 2010
Problems for Recitation 10 Analysis of Two Networks
Two communication networks are shown below. Complete the table of properties and be prepared to justify yo
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
October 8, 2010
Notes for Recitation 9 1 Traveling Salesperson Problem
Now we're going to talk about a famous optimization problem known as the Traveling Sales person Problem1
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
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 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 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 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
September 29, 2010
Notes for Recitation 6 1 Graph Basics
B A F D G C E
Let G = (V, E) be a graph. Here is a picture of a graph.
Recall that the elements of V are called vertic
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 29, 2010
Problems for Recitation 6 1 Graph Basics
B A F D G C E
Let G = (V, E) be a graph. Here is a picture of a graph.
Recall that the elements of V are called ver
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 24, 2010
Notes for Recitation 5 1 Exponentiation and Modular Arithmetic
Recall that RSA encryption and decryption both involve exponentiation. To encrypt a message m
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 24, 2010
Problems for Recitation 5 1 RSA: Let's try it out!
You'll probably need extra paper. Check your work carefully! 1. As a team, go through the beforehand step
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 22, 2010
Notes for Recitation 4 1 The Pulverizer
We saw in lecture that the greatest common divisor (GCD) of two numbers can be written as a linear combination of th
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 22, 2010
Problems for Recitation 4 1 Problem: The Pulverizer!
There is a pond. Inside the pond there are n pebbles, arranged in a cycle. A frog is sitting on one of
6.042/18.062J Mathematics for Computer Science Tom Leighton, Marten van Dijk
September 17, 2010
Notes for Recitation 3 1 State Machines
Recall from Lecture 3 (9/16) that an invariant is a property of a system (in lecture, that system was the 8-puzzle) tha
6.042/18.062J Mathematics for Computer Science Tom Leighton, Marten van Dijk
September 17, 2010
Problems for Recitation 3 1 Problem: Breaking a chocolate bar
We are given a chocolate bar with m n squares of chocolate, and our task is to divide it into mn
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 15, 2010
Notes for Recitation 2 1 Induction
Recall the principle of induction:
Principle of Induction. Let P (n) be a predicate. If
P (0) is true, and for all n N,
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 15, 2010
Problems for Recitation 2 1 Problem: A Geometric Sum
1 - rn+1 1-r
Perhaps you encountered this classic formula in school: 1 + r + r2 + r3 + . . . + rn =
Fir
6.042/18.062J Mathematics for Computer Science Tom Leighton and Marten van Dijk
September 10, 2010
Notes for Recitation 1 1 Logic
How can one discuss mathematics with logical precision, when the English language is itself riddled with ambiguities? For exa
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
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 +
"mcs-ftl" - 2010/9/8 - 0:40 - page 533 - #539
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 497 - #503
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.
"mcs-ftl" - 2010/9/8 - 0:40 - page 467 - #473
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
"mcs-ftl" - 2010/9/8 - 0:40 - page 445 - #451
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
"mcs-ftl" - 2010/9/8 - 0:40 - page 431 - #437
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