Statistical Distributions Book

1 Introduction.

2 Terms and Symbols.

2.1 Probability, Random Variable, Variate and Number.

2.2 Range, Quantile, Probability and Domain.

2.3 Distribution Function and Survival Function.

2.4 Inverse Distribution and Inverse Survival Function.

2.5 Probability Density Function and Probability Function.

2.6 Other Associated Functions and Quantities.

3 General Variate Relationships.

3.1 Introduction.

3.2 Function of a Variate.

3.3 One-to-One Transformations and Inverses.

3.4 Variate Relationships Under One-to-One Transformation.

3.5 Parameters, Variate, and Function Notation.

3.6 Transformation of Location and Scale.

3.7 Transformation from the Rectangular Variate.

3.8 Many-to-One Transformations.

4 Multivariate Distributions.

4.1 Joint Distributions.

4.2 Marginal Distributions.

4.3 Independence.

4.4 Conditional Distributions.

4.5 Bayes' Theorem.

4.6 Functions of a Multivariate.

5 Stochastic Modeling.

5.1 Introduction.

5.2 Independent Variates.

5.3 Mixture Distributions.

5.4 Skew-Symmetric Distributions.

5.5 Conditional Skewness.

5.6 Dependent Variates.

6 Parameter Inference.

6.1 Introduction.

6.2 Method of Percentiles Estimation.

6.3 Method of Moments Estimation.

6.4 Maximum Likelihood Inference.

6.5 Bayesian Inference.

7 Bernoulli Distribution.

7.1 Random Number Generation.

7.2 Curtailed Bernoulli Trial Sequences.

7.3 Urn Sampling Scheme.

7.4 Note.

8 Beta Distribution.

8.1 Notes on Beta and Gamma Functions.

8.2 Variate Relationships.

8.3 Parameter Estimation.

8.4 Random Number Generation.

8.5 Inverted Beta Distribution.

8.6 Noncentral Beta Distribution.

8.7 Beta Binomial Distribution.

9 Binomial Distribution.

9.1 Variate Relationships.

9.2 Parameter Estimation.

9.3 Random Number Generation.

10 Cauchy Distribution.

10.1 Note.

10.2 Variate Relationships.

10.3 Random Number Generation.

10.4 Generalized Form.

11 Chi-Squared Distribution.

11.1 Variate Relationships.

11.2 Random Number Generation.

11.3 Chi Distribution.

12 Chi-Squared (Noncentral) Distribution.

12.1 Variate Relationships.

13 Dirichlet Distribution.

13.1 Variate Relationships.

13.2 Dirichlet Multinomial Distribution.

14 Empirical Distribution Function.

14.1 Estimation from Uncensored Data.

14.2 Estimation from Censored Data.

14.3 Parameter Estimation.

14.4 Example.

14.5 Graphical Method for the Modified Order-Numbers.

14.6 Model Accuracy.

15 Erlang Distribution.

15.1 Variate Relationships.

15.2 Parameter Estimation.

15.3 Random Number Generation.

16 Error Distribution.

16.1 Note.

16.2 Variate Relationships.

17 Exponential Distribution.

17.1 Note.

17.2 Variate Relationships.

17.3 Parameter Estimation.

17.4 Random Number Generation.

18 Exponential Family.

18.1 Members of the Exponential Family.

18.2 Univariate One-Parameter Exponential Family.

18.3 Estimation.

18.4 Generalized Exponential Distributions.

19 Extreme Value (Gumbel) Distribution.

19.1 Note.

19.2 Variate Relationships.

19.3 Parameter Estimation.

19.4 Random Number Generation.

20 F (Variance Ratio) or Fisher{ Snedecor Distribution.

20.1 Variate Relationships.

21 F (Noncentral) Distribution.

21.1 Variate Relationships.

22 Gamma Distribution.

22.1 Variate Relationships.

22.2 Parameter Estimation.

22.3 Random Number Generation.

22.4 Inverted Gamma Distribution.

22.5 Normal Gamma Distribution.

22.6 Generalized Gamma Distribution.

22.6.1 Variate Relationships.

23 Geometric Distribution.

23.1 Notes.

23.2 Variate Relationships.

23.3 Random Number Generation.

24 Hypergeometric Distribution.

24.1 Note.

24.2 Variate Relationships.

24.3 Parameter Estimation.

24.4 Random Number Generation.

24.5 Negative Hypergeometric Distribution.

24.6 Generalized Hypergeometric (Series) Distribution.

25 Inverse Gaussian (Wald) Distribution.

25.1 Variate Relationships.

25.2 Parameter Estimation.

26 Laplace Distribution.

26.1 Variate Relationships.

26.2 Parameter Estimation.

26.3 Random Number Generation.

27 Logarithmic Series Distribution.

27.1 Variate Relationships.

27.2 Parameter Estimation.

28 Logistic Distribution.

28.1 Notes.

28.2 Variate Relationships.

28.3 Parameter Estimation.

28.4 Random Number Generation.

29 Lognormal Distribution.

29.1 Variate Relationships.

29.2 Parameter Estimation.

29.3 Random Number Generation.

30 Multinomial Distribution.

30.1 Variate Relationships.

30.2 Parameter Estimation.

31 Multivariate Normal (Multinormal) Distribution.

31.1 Variate Relationships.

31.2 Parameter Estimation.

32 Negative Binomial Distribution.

32.1 Note.

32.2 Variate Relationships.

32.3 Parameter Estimation.

32.4 Random Number Generation.

33 Normal (Gaussian) Distribution.

33.1 Variate Relationships.

33.2 Parameter Estimation.

33.3 Random Number Generation.

33.4 Truncated Normal Distribution.

33.5 Variate Relationships.

34 Pareto Distribution.

34.1 Note.

34.2 Variate Relationships.

34.3 Parameter Estimation.

34.4 Random Number Generation.

35 Poisson Distribution.

35.1 Note.

35.2 Variate Relationships.

35.3 Parameter Estimation.

35.4 Random Number Generation.

36 Power Function Distribution.

36.1 Variate Relationships.

36.2 Parameter Estimation.

36.3 Random Number Generation.

37 Power Series (Discrete) Distribution.

37.1 Note.

37.2 Variate Relationships.

37.3 Parameter Estimation.

38 Queuing Formulas.

38.1 Characteristics of Queuing Systems.

38.2 Definitions, Notation and Terminology.

38.3 General Formulas.

38.4 Some Standard Queuing Systems.

39 Rayleigh Distribution.

39.1 Variate Relationships.

39.2 Parameter Estimation.

40 Rectangular (Uniform) Continuous Distribution.

40.1Variate Relationships.

40.2 Parameter Estimation.

40.3 Random Number Generation.

41 Rectangular (Uniform) Discrete Distribution.

41.1 General Form.

41.2 Parameter Estimation.

42 Student's t Distribution.

42.1 Variate Relationships.

42.2 Random Number Generation.

43 Student's t (Noncentral) Distribution.

43.1 Variate Relationships.

44 Triangular Distribution.

44.1 Variate Relationships.

44.2 Random Number Generation.

45 von Mises Distribution.

45.1 Note.

45.2 Variate Relationships.

45.3 Parameter Estimation.

46 Weibull Distribution.

46.1 Note.

46.2 Variate Relationships.

46.3 Parameter Estimation.

46.4 Random Number Generation.

46.5 Three-Parameter Weibull Distribution.

46.6Three-Parameter Weibull Random Number Generation.

46.7 Bi-Weibull Distribution.

46.8 Five-Parameter Bi-Weibull Distribution.

Bi-Weibull Random Number Generation.

Bi-Weibull Graphs.

46.9 Weibull Family.

47 Wishart (Central) Distribution.

47.1 Note.

47.2 Variate Relationships.

48 Statistical Tables.

Bibliography.