Part 1. Random variables & noteworthy distributions, basic probability models.
Random variables and their characteristics: mean, moments, probability generating functions, moment generating functions, characteristic functions (cenni). Noteworthy discrete and continuous distributions: Bernoulli, Binomial,uniform, Poisson, Pascal, Normal, exponential, Gamma, Beta.
Random vectors. Functions of Random variables. Random vectors with independent components.
Transformation Tecniques to find the distribution of a function of a given random variable. Sum of independent & equally distributed random variables. Convolutions. The moment generating function technique.
Introduction to probability models. Hazard models: constant hazard and the esponential law, multiple transitions with constant hazard (Gamma), epidemiological risk (logistic distribution), linear risk (Weibull), constant + epidemiologicl risk.
Sequences of random variables. Various notions of Convergence. Law of large numbers. Convergence in distribution and central limit theorem.
Part 2 Inference. Sampling, estimation, statistical testing.
Sampling: basic notions.
The likelihood function and the likelihood principle. Estimation. The maximum likelihood technique. Properties required to “nice” point estimators. Consistency. Sufficiency and accuracy. Cramér-Rao lemma. Properties of maximum likelihood estimators (MGB ch. 7-8).
Interval estimation. The likelihood approach. The pivotal quantity approach. Costruction of interval estimates for scalal and vector parameters for normal laws. Asymptotic intervals based on maximum likelihood estimators. Asymptotic estimates of a bernoulli proportion and related strange phenomena.
Hypothesis testing. The likelihood ratio test. Lemma di Neyman Pearson.
Nonlinear estimation and mixture models: a glance.