1. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.1 Binomial distribution Error of an efficiency Estimator for the efficiency (when k out of n observed): Estimator for the “error” on k note: „n“ has no error note: pathological behaviour for k=0 and k=n, estimated error = 0  „bias“  more on estimators later

2. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution Discrete probability distribution Describes events whose individual probability p is very small while the number of trials N is very large, such that Np is finite Examples: radioactive decay of source (e.g. 10 23 nuclei, only few decays per second) number of raisins in a slice of cake (r(raisin) << r(cake) probability to observe N collisions of a given type in a collider experiment ( N(colliding particles/bunch) >> N(collisions) ) number of lightnings in a thunder strom within 1 minute number of plane crashes per year, … All these examples can in principle be treated by the Binomial Distribution but numerically tedious (factorials of large numbers) Sometime neither N nor p but only Np =: is known (or can be estimated)

3. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution Math. derivation (from Binomial distribution): Poisson distribution:

4. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution

5. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution Alternative (more physical) derivation: - assume an experiment which counts r events in a time T - divide T in small intervals  t, such that in one interval only 0 or 1 events occur Probability for one event in [t,t+  t]:  t Probability for no event in [t,t+  t]: 1-  t Call probability for no event in [0,t] =: P o (t) Poisson for 0 events

6. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution Probability for one event: either in [0,t] or [t,t+  t]

7. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution (properties) Normalization: Expectation value: Variance: follows from variance of Binomial distribution

8. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.2 Poisson distribution (properties) Note: Binomial distribution bounded from above (k <= N) Poisson not bounded from above Not poisson distributed: - number of decays of a radioactive source per intervall T when T ~ o (  ) (lifetime) - observed rate of event in a thick target

9. Gauss (Normal) distribution: Random Variable: x (continuous) Two Parameters: Most important probability distribution! Consistent with Poisson distribution for large Consistent with Binomial distribution for large n Normalization: Expectation value: Variance: 3. Distributions 3.3 Gaussian distribution K. Desch – Statistical methods of data analysis SS10

10. Useful integrals: symmetric: [-1 ,1  ] = 68.27 % [-2 ,2  ] = 95.45 % [-3 ,3  ] = 99.73 % one-sided: [- ,1  ] = 84.13 % [- ,2  ] = 97.72 % [- ,3  ] = 99.87 % 3. Distributions 3.3 Gaussian distribution K. Desch – Statistical methods of data analysis SS10

11. 3. Distributions 3.3 Gaussian distribution: limits Binomial distributionPoisson distribution Normal distribution K. Desch – Statistical methods of data analysis SS10 Proof:

12. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.3 Gaussian distribution Comparison: Gauss and Poisson distributions

13. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.3 Gaussian distribution Two-sided Gaussian integral

14. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.3 Gaussian distribution One-side Gaussian integral

15. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.3 Gaussian distribution Integrated Gauss function Full Width at the Half-Height (FWHH) Gauss error function

16. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.4 Multi-dimensional Gaussian Multidimensional Gauss distribution N Gauss distributed random variables with mean and covariance matrix V ( ). The common p.d.f.: For two dimensions: Area inside 1  -interval: 39.35% (smaller than in 1D case !)

17. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.5 Characteristic function Characteristic function: up to a factor; Fourier transform of f(x) Obtain the original p.d.f. by back transformation: For discrete random variables: Back transformation:

18. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.5 Characteristic function also normal distribution with variance 1/σ 2 For Gauss distribution with arbitrary  : for Gauss distribution with  =0 (normal distribution) phase

19. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.5 Characteristic functions Characteristic function of a sum of random variables is the product of the characteristic functions of these variables: Moments can easily be found from characteristic functions Proof:

20. K. Desch – Statistical methods of data analysis SS10 3. Distributions 3.5 Characteristic functions Sum of two Gauss distributions is also distributed according to a gaussian with: and Proof: