Syllabus
- Basic concepts: probability and the probability measure, a random variable, probability density functions, a random sample, parametrization of probability densities.
- Conditional and marginal probabilities. The Bayes theorem and its use.
- The expectation value and variance of a random variable. Central and non-central moments. Covariance matrix of random variables. Statistical independence. The variance of a function of random variables. Transformations of random variables. The convolution and its properties.
- Characteristic functions of random variables and their use.
- A survey of the most relevant statistical distributions (the uniform, binomial, multinomial, Poisson, normal,chi.square, Student, Fisher, Cauchy, log-normal, etc.). Their properties and situations where we encounter them.
- The Central Limit theorem and an example of it use.
- Estimates of unknown parameters. Consictency and unbiasedness of estimates. Some methods for construction of statistics for estimating parameters. - Likelihood functions and the methd of meaximum likelihood.
- The method opf least squares in its general formulation. The main properties of the weighted quadratic deviation. The Gauss-Markov theorem. The linear model: estimates of parameters, their covatiance matrix, smothing of empirically determined function values, determination of the band of reliability, the problem of numerical stability and its solutions.
- Statistical hypotheses and their testing. The concept of testing statistic. Examples of testing.
Annotation
Basic concepts in probability - random quantities, probability distributions, moments. Parameter estimation by the methods of maximum likelihood and least squares.
Testing of hypotheses. Data processing - analysis of regression, interpolation and extrapolation of data, data reduction, decomposition of spectra.