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October 23. 201-1 MATHEMATICS 121.3 Total Marks - 40
Test#3 Time: 75 minutes
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Contents
III The Mean Value Theorem for Rational Functions III.A III.B III.C III.D Derivatives of Rational Functions and Tangent Lines . . . . . . . . . . . . . . . . . . . . . . Mean Value Theorem, or MVT, and Rolle's Theorem for Rational Functions . . .
Week 2: Conditional Probability and Bayes formula
We ask the following question: suppose we know that a certain event B has occurred.
How does this impact the probability of some other A. This question is addressed by
conditional probabilities. We write
P
Homework 2
Exercise 1: A coin is tossed three times. What is the probability that two heads occur
given that
The rst outcome was a head.
The rst outcome was a tail.
The rst two outcomes were heads.
The rst two outcomes were tails.
The rst outcome was
Week 5: Expected value and Betting systems
Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y, . If S is the sample
space of the experiment then to each i S the ran
Chapter 2: Conditional Probability and Bayes
formula
We ask the following question: suppose we know that a certain event B has occurred.
How does this impact the probability of some other A. This question is addressed by
conditional probabilities. We writ
Week 8: Basic concepts in game theory
Part 1: Examples of games
We introduce here the basic objects involved in game theory. To specify a game ones
gives
The players.
The set of all possible strategies for each player.
The payos: if each player picks a
Homework 1
Exercise 1: As seen in class if there are 23 people in a room the probability of having
two people with the same birthday is more than 1/2. In our class of 42 people what is
this probability? For comparison compute the probability that somebody
Week 4: Gamblers ruin and bold play
Random walk and Gamblers ruin. Imagine a walker moving along a line. At every
unit of time, he makes a step left or right of exactly one of unit. So we can think that his
position is given by an integer n Z. We assume t
Week 6: Variance, the law of large numbers and
Kellys criterion
Expected value, variance, and Chebyshev inequality. If X is a random variable
recall that the expected value of X, E[X] is the average value of X
Expected value of X :
E[X] =
P (X = )
The ex
Week 1: Probability models and counting
Part 1: Probability model
Probability theory is the mathematical toolbox to describe phenomena or experiments
where randomness occur. To have a probability model we need the following ingredients
A sample space S w
Chapter 7: Proportional Play and the Kelly Betting
System
Proportional Play and Kellys criterion: Investing in the stock market is, in eect,
making a series of bets. Contrary to bets in a casino though, one would generally believe
that the stock market is
Week 9.1: Nash equilibrium for monopolies and
duopolies
We discuss here an application of Nash equilibrium in economics, the Cournots
duopoly model. This is a very classical problem which in fact predates modern game
theory by more than a century.
Supply
Week 3, Part 2: Linear dierence equations
In this lecture we discuss how to solve linear dierence equations.
First order homogeneous equation: You should think of the time being discrete and
taking integer values n = 0, 1, 2, and qn describing the state o