1. Overview course in Statistics (usually given in 26h, but now in 2h)  introduction of basic concepts of probability  concepts of parameter estimation and confidence belts  techniques: maximum likelihood & least-square  hypothesis testing & goodness-of-fit concepts Prof. Jorgen D’Hondt (Vrije Universiteit Brussel) IPN (Teheran, Iran) Winter School – February, 2008

2. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)2 It will go fast...

3. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)3 Lecture 1: content Basic concepts of probability theory definition of probability & main calculation rules rule of Bayes permutations & combinations & binomial distribution Probability distributions experimental histograms versus theoretical distributions definition of a distribution of a stochastic variable concept of correlation between stochastic variables Main distributions used in physics binomial, Poisson, Gaussian/normal, exponential,  2, Cauchy convoluting an experimental resolution with a physics distribution Central Limit theorem in action Parameter estimation basic concept & interpretation of the techniques construction of confidence intervals (method of Neyman) properties of estimators (incl. Minimum Variance Bound) Propagation of uncertainties one variable & many variables with a covariance matrix

4. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)4

5. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)5

6. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)6 Rule of Bayes X i : theorie i Y : experimental result

7. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)7 Experiment of Galton Binomial distribution

8. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)8 Experiment of Galton → binomial distribution N=20 P(left)=0.5 N=20 P(left)=0.25 N=50 P(left)=0.5 N=50 P(left)=0.25

9. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)9 Describe your data-set Discrete (Integers) Continuous (Reals) Binning { 5.6354 7.3625 8.1635 9.3634 1.3846 0.2847 1.4763 } ‘Histograms’ ‘N-tuples’ Quantitative via some variables (mean, N, median, standard deviation,  -percentile,...)

10. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)10 Probability Density Distribution (PDF) cumulative

11. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)11

12. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)12 Poisson distribution : example The Michigan-Irvine-Brookhaven experiment has observed on the 23th of February 1987 in total 9 neutrinos in an interval of 10 seconds. This was exactly on the moment when the supernova S1987a was observed by astronomers. They have compared this event with many observations they made with the same experiment. What is the probability that this special event happened in the time period they have measured? Number of neutrinos Number of times observed in 10 seconds Check what a Poissonian distribution would predict. Answer: In the total time they have measured, they would expect 0.0003 events in which they observe 9 neutrinos in a 10 seconds interval. Hence a very small probability that this was simply a fluctuation from a Poisson distribution...

13. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)13 Central Limit theorem at work N=2 N=50 N=5 N=3 N=10 N=20

14. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)14 Limits between distributions

15. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)15 Limits between distributions: example

16. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)16 No limit for the Cauchy distribution

17. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)17 Experiment versus Theory x – mean of our sample  – mean of our parent dist  – S.D. of our parent dist– sample variance of our sample x s   Data Sample (experiment) Probability Distribution (from which data sample was drawn) Latin letters Greek letters Statistics – parameter estimation

18. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)18 Stochastic X Different samples of n measurements estimator  (one value per sample) result of 1 sample

19. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)19 Confidence intervals (Neyman) 11 22 x0x0

20. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)20 End of first lecture

21. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)21 Lecture 2: content Maximum Likelihood technique general method & example properties of the Maximum Likelihood estimator & variance graphical interpretation Least-Square technique general method the linear model as main case study & matrix notation variance of the estimator & graphical interpretation Hypothesis testing general concepts of a hypothesis test & type of errors made Neyman-Pearson optimal test (likelihood ratio method) Goodness-of-fit tests basic concept & confidence level the  2 -test as most important test & the RUN test

22. Statistics, Teheran, February 2008Prof. Jorgen D'Hondt (Vrije Universiteit Brussel)22