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Hand-book on STATISTICAL DISTRIBUTIONS for …

Internal Report SUF PFY/96 01 Stockholm, 11 December 19961st revision, 31 October 1998last modification 10 September 2007 Hand-book onSTATISTICALDISTRIBUTIONS forexperimentalistsbyChristian WalckParticle Physics GroupFysikumUniversity of Random Number Generation..12 Probability Density Introduction.. Moments.. Errors of Moments.. Characteristic Function.. Probability Generating Function.. Cumulants.. Random Number Generation.. Cumulative Technique.. Accept-Reject technique.. Composition Techniques.. Multivariate DISTRIBUTIONS .. Multivariate Moments.

Internal Report SUF–PFY/96–01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modification 10 September 2007 Hand-book on STATISTICAL

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1 Internal Report SUF PFY/96 01 Stockholm, 11 December 19961st revision, 31 October 1998last modification 10 September 2007 Hand-book onSTATISTICALDISTRIBUTIONS forexperimentalistsbyChristian WalckParticle Physics GroupFysikumUniversity of Random Number Generation..12 Probability Density Introduction.. Moments.. Errors of Moments.. Characteristic Function.. Probability Generating Function.. Cumulants.. Random Number Generation.. Cumulative Technique.. Accept-Reject technique.. Composition Techniques.. Multivariate DISTRIBUTIONS .. Multivariate Moments.

2 Errors of Bivariate Moments.. Joint Characteristic Function.. Random Number Generation..113 Bernoulli Introduction.. Relation to Other DISTRIBUTIONS ..124 Beta Introduction.. Derivation of the Beta distribution .. Characteristic Function.. Moments.. Probability Content.. Random Number Generation..155 Binomial Introduction.. Moments.. Probability Generating Function.. Cumulative Function.. Random Number Generation.. Estimation of Parameters.. Probability Content..186 Binormal Introduction.. Conditional Probability Density.. Characteristic Function.

3 Moments.. Box-Muller Transformation.. Probability Content.. Random Number Generation..247 Cauchy Introduction.. Moments.. Normalization.. Characteristic Function.. Location and Scale Parameters.. Breit-Wigner distribution .. Comparison to Other DISTRIBUTIONS .. Truncation.. Sum and Average of Cauchy Variables.. Estimation of the Median.. Estimation of the HWHM.. Random Number Generation.. Physical Picture.. Ratio Between Two Standard Normal Variables..358 Chi-square Introduction.. Moments.. Characteristic Function.. Cumulative Function.

4 Origin of the Chi-square distribution .. Approximations.. Random Number Generation.. Confidence Intervals for the Variance.. Hypothesis Testing.. Probability Content.. Even Number of Degrees of Freedom.. Odd Number of Degrees of Freedom.. Final Algorithm.. Chi distribution ..439 Compound Poisson Introduction.. Branching Process.. Moments.. Probability Generating Function.. Random Number Generation..4610 Double-Exponential Introduction.. Moments.. Characteristic Function.. Cumulative Function.. Random Number Generation..48ii11 Doubly Introduction.

5 Moments.. Cumulative distribution .. Random Number Generation..5012 Doubly Introduction.. Moments.. Cumulative distribution .. Random Number Generation..5213 Error Introduction.. Probability Density Function..5314 Exponential Introduction.. Cumulative Function.. Moments.. Characteristic Function.. Random Number Generation.. Method by von Neumann.. Method by Marsaglia.. Method by Ahrens..5615 Extreme Value Introduction.. Cumulative distribution .. Characteristic Function.. Moments.. Random Number Generation..6016 Introduction.. Relations to Other DISTRIBUTIONS .

6 1/F.. Characteristic Function.. Moments.. F-ratio.. Variance Ratio.. Analysis of Variance.. Calculation of Probability Content.. The Incomplete Beta function.. Final Formul .. Random Number Generation..68iii17 Gamma Introduction.. Derivation of the Gamma distribution .. Moments.. Characteristic Function.. Probability Content.. Random Number Generation.. Erlangian distribution .. General case.. Asymptotic Approximation..7218 Generalized Gamma Introduction.. Cumulative Function.. Moments.. Relation to Other DISTRIBUTIONS ..7419 Geometric Introduction.

7 Moments.. Probability Generating Function.. Random Number Generation..7520 Hyperexponential Introduction.. Moments.. Characteristic Function.. Random Number Generation..7821 Hypergeometric Introduction.. Probability Generating Function.. Moments.. Random Number Generation..8022 Logarithmic Introduction.. Moments.. Probability Generating Function.. Random Number Generation..8223 Logistic Introduction.. Cumulative distribution .. Characteristic Function.. Moments.. Random numbers..85iv24 Log-normal Introduction.. Moments.. Cumulative distribution .

8 Random Number Generation..8725 Maxwell Introduction.. Moments.. Cumulative distribution .. Kinetic Theory.. Random Number Generation..9026 Moyal Introduction.. Normalization.. Characteristic Function.. Moments.. Cumulative distribution .. Random Number Generation..9327 Multinomial Introduction.. Histogram.. Moments.. Probability Generating Function.. Random Number Generation.. Significance Levels.. Equal Group Probabilities..9628 Multinormal Introduction.. Conditional Probability Density.. Probability Content.. Random Number Generation..10029 Negative Binomial Introduction.

9 Moments.. Probability Generating Function.. Relations to Other DISTRIBUTIONS .. Poisson distribution .. Gamma distribution .. Logarithmic distribution .. Branching Process.. Poisson and Gamma DISTRIBUTIONS .. Random Number Generation..107v30 Non-central Introduction.. Derivation of distribution .. Moments.. Cumulative distribution .. Random Number Generation..10931 Non-central Chi-square Introduction.. Characteristic Function.. Moments.. Cumulative distribution .. Approximations.. Random Number Generation..11232 Introduction.. Moments.. Cumulative distribution .

10 Approximations.. Random Number Generation..11533 Introduction.. Derivation of distribution .. Moments.. Cumulative distribution .. Approximation.. Random Number Generation..11834 Normal Introduction.. Moments.. Cumulative Function.. Characteristic Function.. Addition Theorem.. Independence ofxands2.. Probability Content.. Random Number Generation.. Central Limit Theory Approach.. Exact Transformation.. Polar Method.. Trapezoidal Method.. Center-tail method.. Composition-rejection Methods.. Method by Marsaglia.. Histogram Technique.. Ratio of Uniform Deviates.

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