Università Roma Tre

Introduction to combinatorics.

Axioms of probability.

Conditional probability and independence. Bayes' formula.

Discrete random variables: Bernoulli, Binomial, Poisson, geometric, hypergeometric and negative binomial.

Expected value and variance of a random variable.

Continuous random variables. Density and distribution function.
Uniform, exponential, gamma, and normal distributions.

Simulation of random variables.

Independent random variables and joint laws.
Convolutions and the sum of independent random variables.
Poisson process. Extremal values of independent random variables.

Limit theorems. Markov and Chebyshev's inequalities. Law of large numbers. Moment generating functions.
A sketch of the proof of the central limit theorem.

William Feller, An introduction to probability theory and its applications. Vol. 1, 3rd edition. Wiley, N.Y., (1968).