Time: MWF 11-12 pm
Location: Altgeld Hall 447
Instructor: Marius
Junge
Office hours: TBA, Altgeld 363
Course e-mail: math34402@math.uiuc.edu
E-mail of the grader: glin2@uiuc.edu
Adrress problems with grading first to him-I am willing to find a solution
if you two together don't.
Books: We will use
Sheldon Ross: A first course in probabilityRemark: The book contains more material than we will cover in class. The students are invited to build their own mind by reading the presentation in the book which will definitely differ from the presentation in class.
Grading: 30% HW (submission in pairs), 40 % 2 midterm, 25% final, 5% contribution in class
Remark: The students are expected
to share their responsibilities towards the homework and
to discuss the material among themselves. Discussing between students
is an important part of the learning process.
Every student will explain an example or the solution
of a homework in class!!!
Course description:
This course is intended as an introduction to probability theory. This includes many interesting examples, the right definitions and some little easy proofs. Indeed, the theoretical part including proofs are essential in getting a grip on the main concepts.Plan:
A good example for a probability space is a finite collection of outcomes and every outcome has a certain probability (for examples a fixed number of colored balls in a container). The probability for an event is the sum of all the probabilities of all outcomes with a certain property (the sum of probabilities of green balls). This counting procedures require some combinatorics.
I) Combinatorics: Products, permutations, drawing with and without replacement, binomial coefficients
II) What is probability? Drawing balls: possible
outcomes and probabilities, repeating experiments. What happens for many
repetitions?
Definition of a probability space, the sigma algebra
of events and the probability function. Simple and less simple properties
(additivity, monotony, union of events, probability of increasing unions
and intersections). Examples.
III) Conditional probability and independence: Definition,,
imposing extra evidence, Bayes formula, gold and silver drawer, Polya's
urn model. Definition of independence, example for mutual independent not
independent events, continuous example.
Examples with crimes and industry.
IV) Random variables: Definition, example:
random time of a complete set,
Discrete random variables: binomial (+/-) (Banach match), hypergeometric,
poisson distribution.
expected value, transformation, Markov-Chebychev inequality, Borel-Cantelli
lemma, variance and correlation.
Independent random variables, infinite sequence of Bernoulli
trials, strong law of large numbers (using finite variance).
V) Continuous random variables: Expectation and Variance, transformation, proposition 2.1, uniform random variable, normal random variable, approximation of the binomial distribution, gamma distribution
VI) Joint distribution: Definition, sums of independent random variables, examples, normal, gamma, multivariate distributions, conditional expectation, calculating expectations by conditioning.
VII) Limit theorems: Formulation and illustrations,
sketch of proofs.
Homework