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Master of Science in Astrophysics (P2.2)

 Statistical methods - an introduction (summer semester 2018)

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The following files (pdf) comprise the current lectures notes

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part 1

Chap. 1 Description of data
data types, binning, bar charts, histograms; averages, variance and standard deviation of a data-set

Chap. 2 Probabilities
axioms, approaches, Bayes-theorem

Chap. 3 Probability distributions functions -- one random variable
discrete and continuous distributions, expectation value, central moments, variance, Tchebychev’s inequality, moments, characteristic function, cumulants, variable transformation

Chap. 4 Distributions of several random variables -- multivariate p.d.f.s
marginal distributions,  convolution, moments, covariance and correlation, variable transformation, variable reduction, distributions with more than two variables

Chap. 5 Important distributions and the CLT
binomial, multinomial, Poisson, uniform, normal, binormal, chi-squared, log-normal, law of big numbers, central limit theorem (CLT)

part 2

Chap. 6 Errors 
Measurement errors, error propagation, systematic errors

Chap. 7 Estimation
random sampling, estimators (bias, consistency, efficiency), basic estimators, stratified sampling, finite populations, likelihood (quotient and function), maximum likelihood (ML-) estimators, information inequality and minimum variance bound, minimum variance estimators, asymptotic properties of ML-estimators, errors on ML-estimators, covariances

Chap. 8 Least squares
chi-squared minimization, fitting to a straight line (“linear regression”), variances and correlation, binned data, goodness of fit, errors on x and y

Chap. 9 MCMC (Markov Chain Monte Carlo) -- sampling the posterior
How to obtain the distribution and errors of model parameters given a set of measurements, and a corresponding example

Chap. 10 Confidence intervals and hypothesis testing
confidence intervals (classical and based on likelihood function), errors of first and 2nd kind, significance, power, F-test, Student-test, Likelihood-ratio test, χ2-test, Kolmogorov-Smirnov test, Spearman rank-test, Wilcoxon rank-sum test (Mann-Whitney U-test)