 # Statistics and Data Analysis

General Introduction:

-      Random variables, probability density functions, the Central Limit theorem, cumulative function, properties of estimators, examples and applications.

-          Methods of minimum squares and maximum likelihood, covariance matrix. Applications and examples.

-          Error propagation: some examples and practical applications.

-          Probability theory, Kolmogorov axioms, theorem of Bayes, practical applications.

-          Lemma of Neyman-Pearson. Probability ordering.

-          Interval estimation, confidence intervals, hypothesis testing and p-values, goodness of fit and practical applications. Construction of the power-curve. Coverage for the confidence intervals from maximum likelihood.

-          The problem of the measurement of 0 or very few events. The method of Feldman-Cousins.

-          Technicalities in the generation of random numbers. Simulations of several functional relations.

-          Processes of Markov. Sketch of Markov chain. The process of Filtering and Smoothing. The Kalman filter.

Statistics in HEP:

-          Evaluation of p-values for counting experiments, with and without nuisances.

-          Definition and computation of significance for a signal.

-          Correspondence between p-value and significance in case of non-Gaussian nuisances.

-          Look-elsewhere effect and approximate methods for its estimation.

-          The CLS method and its application to the search for signals.

-          Profile likelihood and statistical tests.

-          Application to the search for the Higgs boson at LHC.

-          Asymptotic methods for the evaluation of sensitivity with the profile likelihood.

-          Use of the Feldman-Cousins method for exclusion plots.

Statistics in Astrophysics:

-          Applications of statistical inference and test of models: Z-score and T-score

-          Coefficient of correlation and related test. Bootstrapping.

-          Non-parametric tests: Spearman’s rank.

-          Kolmogorov-Smirnov: test and related applications, test of Cramér-von Mises

-          Test of isotropy: monolope, dipole and quadrupole, statistics of Rayleigh, Watson and Bingham.

-          Correction of Bonferroni or trial factors.

-          Test of Anderson-Darling.

-          Statistics of Cash (Poisson)

-          Application of maximum likelihood: the catalogue.

-          Errors of type I and type II: screening and testing, technicalities, sensitivity and power of testing.

-          Data analysis: correlation, auto-correlation, function of angular correlation at 2 points, and applications.

-          Analysis of images: linear filters and applications, the Gaussian filter.