MATHEMATICAL AND STATISTICAL METHODS

To provide basic knowledge of statistics, probability, Monte carlo method and Markov chains.

Descriptive statistics. numerical statistical data representations. graphic representations of frequency distributions. indices of central tendency, variability and shape. linear and non-linear regression for a series of data. Exercises with spreadsheet.

Elements of probability. Some definitions of probability. Axiomatic definition of probability. Conditional probability. Bayes' theorem. Discrete and continuous random variables. Central tendency and variability indices.

Important distributions. Bernoulli, Binomial, Poisson, exponential, Weibull, Normal, Chi-square, Student. Convergence theorems. Convergence in distribution, law of large numbers, central limit theorem.

Parameter Estimates. Sampling and samples. Main sampling distributions. Estimators and point estimates. interval estimates: confidence intervals for the mean and variance. Examples

Hypothesis testing. General characteristics of a hypothesis test. Parametric Test. Examples. Nonparametric tests. Test for goodness of fit. Kolmogorov-Smirnov test. Test of Chi-Square. Exercises with spreadsheet.

Random number generation. Generators based on linear recurrences. Statistical tests for random numbers. Generation random numbers with assigned probability density: direct technique, rejection, combined.

Monte Carlo method. Recalls on numerical integration methods. Algorithm Monte Carlo "Hit or Miss". Algorithm Monte Carlo sampling. Algorithm Monte Carlo sample-mean. Techniques of variance reduction: importance sampling, control variates, stratified sampling, antithetic variates. Direct Simulation Monte Carlo for semiconductors.

Markov chains. Definitions and generalities. Joint laws calculus. Classification of states. Probability invariant. Steady state. Metropolis algorithm. Note on queuing theory.

notes provided by the teacher