Università Roma Tre

Combinatorics, assioms of probability, conditional probability and independence, discrete random variables. Continuous random variables, density and distribution functions. Independence and joint laws. Relation between exponential distribution and Poisson distribution. Limit theorems, moment generating functions and sketch of central limit theorem.

- S. Ross, Calcolo delle probabilita' (Apogeo Ed.)
- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)

Combinatorial Analysis. Introduction to combinatorial calculations: permutations,
combinations, examples.

Probability Axioms. Sample spaces, events, probability axioms. Equally likely events and other examples.

Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.

Discrete Random Variables. Bernoulli, binomial, and Poisson random variables.
Poisson process. Other discrete distributions: geometric, hypergeometric, negative binomial. Expected value and variance of a discrete random variable. Examples.

Continuous Random Variables. Probability density function and distribution function. Uniform distribution on an interval, exponential, gamma, normal, Weibull, Cauchy distributions. Link between gamma distributions and Poisson process. Expected value and variance for continuous random variables.

Joint Distributions and Independent Random Variables. Joint distributions, independent random variables. Density of the sum of two independent random variables. Convolution product for normal, gamma, Poisson distributions. Maximum and minimum of independent random variables.

Limit Theorems. Markov's and Chebyshev's inequalities. Weak law of large numbers. Moment generating function and a brief proof of the Central Limit Theorem.

- S. Ross, Calcolo delle probabilita' (Apogeo Ed.)
- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
- W. Feller, An introduction to probability theory and its applications (Wiley, 1968).