Law of Large Numbers
10/27/2005
An intuitive way to view the probability of a certain outcome is the frequency with which that outcome occurs in the long run. We dened probability mathematically as a value of a distribution function for the random variab
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Lecture 13
Math 20 Fall 2014, Dartmouth College
1 Variance of discrete random variables
standard deviation when we talked about the normal density. It was a measure of how
We have already encountered
much the distribution deviates from the mean. It is n
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Lecture 14
Math 20 Fall 2014, Dartmouth College
1 Sums of independent discrete random variables
Let X, Y be independent, integer valued random variables with distributions mX (k) and mY (k) respectively. Let
Z = X + Y. What is the distribution mZ (k)?
W
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Lecture 17
Math 20 Fall 2014, Dartmouth College
In nature, many things - like height and weight of a human being - are distributed according to a normal distribution.
There are many factors that contribute to its value, and the more complex something is
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Lecture 15, 10/20
Math 20 Fall 2014, Dartmouth College
1 Independent trials
Let Sn be the sum of n Bernoulli trials X1 , . . . , Xn , each with p = 1/6.
1. Find = E (Xi ) and 2 = V (Xi ).
2. For any n, nd E (Sn ) and V (Sn ).
3. Find E ( Sn ) and V ( Sn
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Lecture 23
Math 20 Fall 2014, Dartmouth College
1 Unpredictable walks
Your coding assignment this week was to make a simple game, that takes an input of an integer from -10 to 10 and then
simulates a 10-step simple random walk on the integers starting a
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MATH 20 FALL 14 ASSIGNMENT 2
Do each of the following exercises. Hand in the solutions to the book problems on paper at the beginning
of class on Friday, 9/26. Send the code for 2 and 3 by email by the same time. Section 4 contains some
Python informati
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Lecture 26: Probability in Cryptography
Math 20 Fall 2014, Dartmouth College
Contents
1
2
3
4
Public Key Cryptography . . . . . . .
Checksums and the Birthday Problem
Pseudo Random Number Generators .
Code breaking by Metropolis-Hastings
.
.
.
.
.
.
.
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Final Revision Checklist
Math 20 Fall 2014, Dartmouth College
1. Basics
Randomness
Denition of random variable, sample space, event
Discrete
Distribution function m
Tree diagrams
Uniform distribution
Geometric distribution
Continuous
Continuous
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MATH 20 FALL 14 ASSIGNMENT 8, DUE Friday 11/7
The usual: you're encouraged to work with other students, hand in solutions to the problems on Friday
at the beginning of class, send the code in by email by the same time.
1 Joint Distributions and Random W
Practice Midterm Exam
Math 20
October 2014
Name:
Please read the following instructions before starting the exam:
This exam is closed book. You may not give or receive any help during the exam,
though you may ask the instructors for clarication if necess
Practice Midterm Exam
Math 20
October 2014
Name:
Please read the following instructions before starting the exam:
This exam is closed book. You may not give or receive any help during the exam,
though you may ask the instructors for clarication if necess
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MATH 20 FALL 14 ASSIGNMENT 7, DUE FRIDAY 10/31
You're encouraged to discuss these problems with other students in the class. Hand in the solutions on
paper at the beginning of class on Friday, 11/14. Also, be careful - the book uses old names for some
o
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MATH 20 F
ALL 14 ASSIGNMENT 3, DUE FRIDAY 10/3
You're encouraged to discuss these problems with other students in the class. Hand in the solutions to
the book problems and part 2 on paper at the beginning of class on Friday, 10/3. Send the code for part
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Lecture 25
Math 20 Fall 2014, Dartmouth College
1 Markov Chain Monte Carlo
Early in the course, we used a Monte Carlo simulation to estimate . That was pretty cool, who knew is really what
people always told you it was and they weren't lying all along.
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Lecture 16: Puzzles
1 Roulette
Roulette in Vegas works as follows: there are numbers from
1 to 36, colored either black or red, and two extra slots numbered 0 or 00. You can bet on any slot you want, or some of
the properties of the numbers. If you bet
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MATH 20 FALL 14 ASSIGNMENT 5, DUE FRIDAY 10/17
You're encouraged to discuss these problems with other students in the class. Hand in the solutions
to the book problems and your guess for what the general formula is for the expected position of the
small
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LECTURE 1 SUMMARY
1 Probability and Randomness
If we toss a fair coin, it has probability" 1/2 of coming up heads, and probability" 1/2 of coming up tails. This means
that if we toss the coin many times, it will come up heads about half the time. In fac
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LECTURE 3 NOTES
Math 20 Fall 2014, Dartmouth College
1 Innities
The material in the Innities" section will not appear on exams for Math 20, just make sure that you
know what countable" means. But it is one of the most fascinating curiosities in mathemat
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Lecture 18
Math 20 Fall 2014, Dartmouth College
1 Condence Interval
p
of a population supports a
reform. They can't ask eeryone in the population, so they would like to estimate the number
n of people that is sucient
Suppose that a polling agency wants