 Thursday 1/16 Introduction:
Bertrands paradox
asks for a solid foundation of probability theory. The
Petersburg Casino
will be treated later. We will
flip coins more often in this course.
 Tuesday 1/21 Kolmogorov's
Axioms of Probability theory .
Basic properties. Examples: 2 dices,
Boys/Girl problem Bertrand solved.
Homework 1 (PS file)
 Thursday 1/23 Examples: radioactive decay,
Monty Hall problem Continuity property of P,
Conditional probability:
boy or girl problem revisited. An urn problem.
 Tuesday 1/28 The urn problem, Bayes rule,
the drawer problem, independence of events.
Pairwise independence is not enough.
Homework 2 (PS file)
 Thursday 1/30 More examples on independence and conditional probability
and Bayes rule. Theory: Bernoulli formula and Sylvester's
"switchon switch off" formula for the probability of
a union of events.
Reminder: even without probability some questions
can be difficult:
Life also can be complicated
 Tuesday 2/3 Review of
combinatorics
Permutations, Sampling and Combinations. Examples, the
Bursday paradox. Distribution of
Homework 3 (PS file)
 Thursday 2/5 More examples in combinatorics. Especially
the odds in Arizona lotto and some combinatorial questions
in music .
 Tuesday 2/11 Discrete random variables,
densities, distributions, examples of distributions.
How to get onto mars
using probability.
Homework 4 (PS file)
 Thursday 2/12 More examples of
distributions :
A sailor, Euler and
a devilish random variable . Independent random
variables.
 Tuesday 2/18 On distributions: what happens when adding independent
random variables. From Bernoulli to Poisson. Applications.
Checklist for Midterm.
distributions II :
Homework 5 (PS file)
 Thursday 2/20 Midterm topics: Material until Tuesday 2/18.
 Midterm 1 (PS file)
 Tuesday 2/25 Review over Midterm. Expectation of random
discrete variables: Definition, computing examples.
Expectation overview .
Homework 6 (PS file)
 Thursday 2/27 Properties of expectation,
more examples. Definition of variance.
Probability generating functions and its use for
computing expectation and variance.
 Tuesday 3/4 Covariance, correlation,
correlation coefficient, regression line.
Independent random variables are uncorrelated.
Schwartz inequality.
Homework 7 (PS file)
 Thursday 3/6 Excess and other higher moments, sum of independent
random variables. Estimating probabilities with the
Chebychev inequality.
The weak law of large numbers for IID random variables.
 Tuesday 3/11 An application of the weak law of large numbers in analysis:
the Weierstrass theorem
We reviewed some material in form a
Selftest (PS file)
which is corrected by yourself and will not be part
of the grade. The test should be useful for spotting
eventual white spots in the first half of the course.
Homework 8 (PS file)
 Thursday 3/13 Preparation for continuous random variables. Countable
versus uncountable. Why did we introduce sigma algebras?
The
BanachTarski paradox.
What is integration ?
Hacking the code at the foundations of probability
theory with the
theorem of Caratheodory.
The paradox of Schwartz on triangulations of surfaces.
 SPRING BREAK.


