FunktionWin

Graphical MS Windows user interface for ConsoleApp_DistributionFunctions (Schrausser, 2024b).

The following functions were realized, German notation:

Wahrscheinlichkeits-Verteilung (probability distribution)

• Binomial-Funktion $f(X=k|n)$
• Poisson-Funktion $f(X=k|n,p)$
• Geometrische-Funktion $f(X=r|p)$
• Hypergeometrische-Funktion $f(X=k|n,K,N)$
• Exakt binomialer 2-Felder Test $f(X=b|b,c)$
• Exakt hypergeometrischer 4-Felder Test $f(X=a|a,b,c,d)$, Fisher Exact (Fisher, 1922, 1954; s. Agresti, 1992).

Screenshots probability distribution (Fig. 1 - Fig. 5):

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

The fundamental binomial distribution was derived by Bernoulli (1713), s. Schneider (2005a) and above all de Moivre (1711, 1718) with the discovery of the first instance of central limit theorem, to approximate the binomial distribution with the normal distribution, further developed by Gauss (1809, 1823), see Hahn (1970), Hald (1990) or Schneider (2005b).

Theta-Verteilung $\theta$ (characteristic value or $\theta$ distribution)

• $z$-Dichte Funktion $f(x=z)$
• $z$-Funktion $F(x=z)$
• $t$-Funktion $F(x=t)$
• $\chi^2$- Funktion $F(x=\chi²)$
• $F$-Funktion $F(x=F)$
• Effekt-Stärke $\epsilon$, Cohen (1977).

Screenshots $\epsilon$ (Fig. 6, Fig. 7):

Figure 6.

Figure 7.

The $t$-distribution was first derived by Lüroth (1876), later in a more general form defined as Pearson Type IV (Pearson, 1895), commonly known as Student's $t$-distribution, from William Sealy Gosset (1908).

Helmert (1876) first described the $\chi^2$-distribution, independently rediscovered by Pearson (1900), c.f. also Elderton (1902), Pearson (1914) or Plackett (1983), for the F-distribution by Fisher (1924) see Snedecor (1934) and Scheffé (1959).

Statistical power $1-\beta$ and effect size $\epsilon$ (Cohen, 1977, 1992) layed foundations for statistical meta-analysis and methods of estimation statistics, see e.g. Borenstein et al. (2001) for related software applications.

Transformationen (transformation functions)

• Fisher $Z$ Funktion $F(x=r)$, Fisher (1915).
• Gamma $F(x)=\Gamma$

Gamma $\Gamma$, to solve the problem of extending the factorial to non-integer arguments, was first considered in a letter from Bernoulli to Goldbach (Bernoulli, 1729), introduced later by Euler (1738) - of fundamental definitional importance for the formulation of approximate probability distribution functions such as $\chi^2$, $t$ or $F$ (c.f. Meyberg and Vachenauer, 2001; Cuyt et al., 2008; Beals and Wong, 2020; Little et al., 2022).

See further e.g. Bortz (1984), Bortz and Weber (2005), Bortz and Schuster (2010), Döring (2023), Pascucci (2024a, b) and Schrausser (2024a).

References

Agresti, A. (1992). A Survey of Exact Inference for Contingency Tables. Statistical Science 7 (1): 131–53. https://doi.org/10.1214/ss/1177011454

Beals, R., & Wong, R. S. C. (2020). The Gamma and Beta Functions. In Explorations in Complex Functions, 141–53. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-54533-8_10

Bernoulli, D. (1729). Lettre XLVII. D. Bernoulli a Goldbach. St.-Petersbourg ce 6. octobre 1729. https://commons.m.wikimedia.org/wiki/File:DanielBernoulliLetterToGoldbach-1729-10-06.jpg

Bernoulli, J. (1713). Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis. Basileae: Impensis Thurnisiorum, Fratrum. https://www.e-rara.ch/zut/doi/10.3931/e-rara-9001

Borenstein, M., Rothstein, H., Cohen, J., Schoenfeld, D., Berlin, J., & Lakatos, E. (2001). Power and Precision: A Computer Program for Statistical Power Analysis and Confidence Intervals. Englewood, NJ: Biostat, Inc. https://books.google.com/books?id=tYg02XZBeNAC&printsec=frontcover&hl=de#v=onepage&q&f=false

Bortz, J. (1984). Lehrbuch Der Empirischen Forschung. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-00468-5

Bortz, J., & Schuster, C. (2010). Statistik Für Human- Und Sozialwissenschaftler: Limitierte Sonderausgabe. 7th ed. Springer-Lehrbuch. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-12770-0

Bortz, J., & Weber, R. (2005). Statistik: Für Human- Und Sozialwissenschaftler. 6th ed. Springer-Lehrbuch. Berlin, Heidelberg: Springer. https://doi.org/10.1007/b137571

Cohen, J. (1977). Statistical Power Analysis for the Behavioral Science. Amsterdam: Elsevier Academic Press. https://doi.org/10.1016/C2013-0-10517-X

———. (1992). A Power Primer. Psychological Bulletin 112 (1): 155–59. https://doi.org/10.1037/0033-2909.112.1.15

Cuyt, A., Petersen, V. B., Verdonk, B., Waadeland, H., & Jones, W. B. (2008). Gamma Function and Related Functions. In Handbook of Continued Fractions for Special Functions, 221–51. Dordrecht: Springer Netherlands. https://doi.org/10.1007/978-1-4020-6949-9_12

de Moivre, A. (1711). De mensura sortis, seu, de probabilitate eventuum in ludis a casu fortuito pendentibus. Philosophical Transactions of the Royal Society of London 27 (329): 213–64. https://doi.org/10.1098/rstl.1710.0018

———. (1718). The Doctrine of Chances: Or, A Method of Calculating the Probability of Events in Play. 1st ed. London: W. Pearson. https://books.google.com/books?id=3EPac6QpbuMC

Döring, N. (2023). Forschungsmethoden Und Evaluation in Den Sozial- Und Humanwissenschaften. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-64762-2

Elderton, W. P. (1902). Tables for Testing the Goodness of Fit of Theory to Observation. Biometrika 1 (2): 155–63. https://doi.org/10.1093/biomet/1.2.155

Euler, L. (1738). De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt. Commentarii Academiae Scientiarum Petropolitanae 5: 36–57. https://scholarlycommons.pacific.edu/euler-works/19/

Fisher, R. A. (1915). Frequency Distribution of the Values of the Correlation Coefficient in Samples from an Indefinitely Large Population. Biometrika 10 (4): 507–21. https://doi.org/10.2307/2331838

———. (1922). On the Interpretation of χ2 from Contingency Tables, and the Calculation of p. Journal of the Royal Statistical Society 85 (1): 87–94. https://doi.org/10.2307/2340521

———. (1924). On a Distribution Yielding the Error Functions of Several Well-Known Statistics. Proceedings International Mathematical Congress, Toronto 2: 805–13. https://repository.rothamsted.ac.uk/item/8w2q9/on-a-distribution-yielding-the-error-functions-of-several-well-known-statistics

———. (1954). Statistical Methods for Research Workers. 12th ed. Edinburgh: Oliver; Boyd. https://www.worldcat.org/de/title/statistical-methods-for-research-workers/oclc/312138

Gauss, C. F. (1809). Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm. Hambvrgi: Svmtibvs F. Perthes et I. H. Besser. https://archive.org/details/theoriamotuscor00gausgoog/page/n1/mode/1up

———. (1823). Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. Göttingen: apud Henricum Dieterich. https://doi.org/10.3931/e-rara-2857

Gosset, W. S. (1908). The Probable Error of a Mean. Biometrika 6 (1): 1–25. https://doi.org/10.2307/2331554

Hahn, R. (1970). Mathematics - The Doctrine of Chances or, A Method of Calculating the Probabilities of Events in Play. By Abraham de Moivre. 2nd ed. [1738]. London, F. Cass, 1967. Pp. xiv + 258. £6 6s. The British Journal for the History of Science 5 (2): 189–90. https://doi.org/10.1017/S0007087400010967

Hald, A. (1990). De Moivre and the Doctrine of Chances, 1718, 1738, and 1756. In History of Probability and Statistics and Their Applications before 1750, edited by Hald, A., 397–424. New York: Wiley Series in Probability; Statistics, Wiley-Interscience. https://onlinelibrary.wiley.com/doi/book/10.1002/0471725161

Helmert, F. R. (1876). Ueber Die Wahrscheinlichkeit Der Potenzsummen Der Beobachtungsfehler Und Über Einige Damit Im Zusammenhange Stehende Fragen. Zeitschrift Für Mathematik Und Physik 21: 192–219. https://gdz.sub.uni-goettingen.de/id/PPN599415665_0021

Little, C. H. C., Teo, K. L., & van Brunt, B. (2022). The Gamma Function. In An Introduction to Infinite Products, 131–91. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-90646-7_3

Lüroth, J. (1876). Vergleichung von Zwei Werthen Des Wahrscheinlichen Fehlers. Astronomische Nachrichten 87 (14): 209–20. https://doi.org/10.1002/asna.18760871402

Meyberg, K., & Vachenauer, P. (2001). Höhere Mathematik 1: Differential- und Integralrechnung Vektor- und Matrizenrechnung. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-56654-7

Pascucci, A. (2024a). Probability Theory I. Random Variables and Distributions. 1st ed. UNITEXT. Cham: Springer. https://doi.org/10.1007/978-3-031-63190-0

———. (2024b). Probability Theory II. Stochastic Calculus. 1st ed. UNITEXT. Cham: Springer. https://doi.org/10.1007/978-3-031-63193-1

Pearson, K. (1895). Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 186: 343–414. https://doi.org/10.1098/rsta.1895.0010

———. (1900). X. On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 50 (302): 157–75. https://doi.org/10.1080/14786440009463897

———. (1914). On the Probability That Two Independent Distributions of Frequency Are Really Samples of the Same Population, with Special Reference to Recent Work on the Identity of Trypanosome Strains. Biometrika 10: 85–154. https://doi.org/10.1093/biomet/10.1.85

Plackett, R. L. (1983). Karl Pearson and the Chi-Squared Test. International Statistical Review / Revue Internationale de Statistique 51 (1): 59–72. https://doi.org/10.2307/1402731

Scheffé, H. (1959). The Analysis of Variance. New York: Wiley. https://psycnet.apa.org/record/1961-00074-000

Schneider, I. (2005a). Chapter 6 - Jakob Bernoulli, Ars conjectandi (1713). In Landmark Writings in Western Mathematics 1640-1940, edited by Grattan-Guinness, I., Cooke, R., Corry, L., Crépel, P., & Guicciardini, N., 88–104. Amsterdam: Elsevier Science. https://doi.org/10.1016/B978-044450871-3/50087-5

———. (2005b). Chapter 7 -Abraham De Moivre, The Doctrine of Chances (1718, 1738, 1756). In Landmark Writings in Western Mathematics 1640-1940, edited by Grattan-Guinness, I., Cooke, R., Corry, L., Crépel, P., & Guicciardini, N., 105–20. Amsterdam: Elsevier Science. https://doi.org/10.1016/B978-044450871-3/50087-5

Schrausser, D. G. (2024a). Handbook: Distribution Functions (Verteilungs Funktionen). PsyArXiv. https://doi.org/10.31234/osf.io/rvzxa

———. (2024b). Schrausser/ConsoleApp_DistributionFunctions: Console applicationes for distribution functions (version v1.5.0). Zenodo. https://doi.org/10.5281/zenodo.7664141

Snedecor, G. W. (1934). Calculation and Interpretation of Analysis of Variance and Covariance. Ames, Iowa: Collegiate Press. https://doi.org/10.1037/13308-000