Statistical analysis of data: course outline

Main topics (approx. 10 lectures):

Probability. Definition and interpretation, random variables, probability density functions, expectation values, transformation of variables, error propagation.

Examples of probability functions. Binomial, multinomial, Poisson, uniform, exponential, Gaussian, chi-square, Cauchy distributions.

The Monte Carlo method. Random number generators, the transformation method, the acceptance-rejection method.

Statistical tests. Significance and power of a test, choice of the critical region. Constructing test statistics: the Fisher discriminant, neural networks. Testing goodness-of-fit, chi^2-test, P-values.

Parameter estimation: general concepts. Samples, estimators, bias. Estimators for mean, variance, covariance.

The method of maximum likelihood. The likelihood function, ML estimators for parameters of Gaussian and exponential distributions. Variance of ML estimators, the information inequality, extended ML, ML with binned data.

The method of least squares. Relation to maximum likelihood, linear least squares fit, LS with binned data, testing goodness-of-fit, combining measurements with least squares.

Interval estimation. Classical confidence intervals: with Gaussian distributed estimator, for mean of Poisson variable. Setting limits, limits near a physical boundary.

Optional topics:

Overview of Bayesian statistics. The Bayesian method: subjective probability and Bayes' theorem. Relation of ML and Bayesian parameter estimation. Bayesian limits.

Characteristic functions. Definition and properties, use to obtain moments, use of addition theorem, deriving limiting cases of distributions, the central limit theorem.

Unfolding. Formulation of the problem: response function and matrix. Inversion of the response matrix, bin-by-bin correction factors. Regularized unfolding: regularization functions, bias and variance of estimators, choice of regularization parameter.