Probability Theory  PRO

Instructor: Dr. Tamás SZABADOS

Text: W. Feller: An Introduction to  Probability Theory and  its  Applications  +  handouts

Prerequisite: elementary calculus


Boolean  operations,  algebra of  events,  probabilities.  (F/I=Feller,  ch.  I.)
Inclusion-exclusion:  the  Sieve Formula.  (F/IV.1)
Combinatorial  reasoning in  determining  probabilities (the  hypergeometric distribution,  occupancy  problems). (F/II,IV)
Conditional probability, Bayes' rule,  independence. (F/V)
Geometrical  reasoning in  determining  probabilities (some  problems). (Handout)
Discrete  random  variables  and their distributions  (hypergeometric, binomial,  geometric  and Poisson  distributions). (F/VI)
Poisson  approximation of  binomial distribution. (F/VI)
The  expectation,  variance and covariance  of discrete random variables.  (F/IX)
Nonnegative integer valued  random variables. Convolution and  the generating  function. (F/XI)
Applications of the generating function: branching process,  first passage and recurrence problems for random walks.  (F/XII)
Bernoulli's Law of Large Numbers for binomially distributed random variables. (F/VI)
An application:  a probabilistic  proof of Weierstrass' approximation theorem. (Handout)
Normal  approximation of the binomial distribution.  (F/VII)
The general notion of  random variable,  distribution  functions (examples: uniform,  exponential, normal (or  Gauss), Cauchy and lognormal  distributions).
Convolution. Convolution of Gaussians: stability.  Convolution of  exponentials: the $\Gamma$-distributions. (Handout)
Expectation,  variance,  covariance  and higher moments of general random variables.  (Handout)
Markov's and Chebyshev's inequalities.  The Weak Law of Large Numbers.  (F/IX,X)
The Borel--Cantelli  lemma and  the  Strong  Law of Large Numbers.  (F/VIII)
Kolmogorov's inequality  and  the Strong Law of Large Numbers (continued).  (F/IX,X)
The  characteristic function,  basic  properties,  examples. (Handout)
Weak convergence of  probability distributions, equivalent formulations.  (Handout)
The method  of  characteristic  functions  in  proving weak convergence:  the Central  Limit  Theorem. (Handout)

With special emphasis on problem solving!