Probability Theory

Historical introduction. Basic concepts: random experiment, event space, event, elementary event, operation between events, axioms, sigma algebra

Properties of probability: Poincare-rule, Boole’s inequalities, continuity of probability

Conditonial probability, independency of events, Theorem of total probability, Bayes’s Theorem, produce theory

Classical probability, geometrical probability. Examples for application: urn models, Buffon’s needle

Random variable, probability distribution function, discrete and continouos cases, properties of the distribution function, probability of falling in an interval, discrete distribution, probability density function

Notable discrete random variables: binomial, Poisson, geometrical.  Poisson approximation to the binomial distribution. Memoryless properties of the geometric distribution

Notable continouos distributions: uniform, exponential, normal. Simulation with uniform distribution. Memoriless property of the exponential distribution. Standard normal distribution. Linear transformation

Expected value, deviation, moments.  Theorems for expected value and deviation. Expected value and deviation of notable dispersions.

Steiner’s Theorem, Markov’s- and Chebisev’s inequalities.

Joint distribution function, projective distribution functions. Independency, convolution (discrete and continouos cases), Joint density function, projective density function

Theorems of large numbers: Weak- and Stronge Law of Large Numbers. Central limit theorems, Moivre-Laplace’s Theorem

Covariance, correlation. Properties of covariance and correlations. Connection between independency and uncorrelatedness

Conditional distribution, conditional expectation (regression). Linear regression. Properties of regression. Examples of discrete and continouos cases

Two-dimensional normal distribution,  polynomial distribution. Connection of the independency and uncorrelatedness in normal case. Projections of the polinomial distribution are binomials