FunktionWin: Windows Interface for distribution functions

Schrausser, D. G. (2025). FunktionWin: Graphical MS Windows user interface for ConsoleApp_DistributionFunctions. FunktionWin DOI:10.5281/zenodo.7651660 Dietmar G. Schrausser orcid.org/0000-0002-4924-8280 Karl-Franzens University, Graz, Austria Graphical MS Windows user interface for ConsoleApp_DistributionFunctions (Schrausser, 2024b). The following functions were realized, German notation: 1. 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). 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). 2. Theta-Verteilung 𝜽 (characteristic value or 𝜽 distribution) • 𝑧-Dichte Funktion 𝑓(𝑥 = 𝑧). • 𝑧-Funktion 𝐹(𝑥 = 𝑧). • 𝑡-Funktion 𝐹(𝑥 = 𝑡). • 𝜒2Funktion 𝐹(𝑥 = 𝜒²). • 𝐹-Funktion 𝐹(𝑥 = 𝐹). • Effekt-Stärke 𝜖, Cohen (1977). 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 𝜒2-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 − 𝛽 and effect size 𝜖 (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. 3. Transformationen (transformation functions) • Fisher 𝑍 Funktion 𝐹(𝑥 = 𝑟), Fisher (1915). • Gamma 𝐹(𝑥)= 𝛤. Gamma 𝛤, to solve the problem of extending the factorial to noninteger 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 𝜒2, 𝑡 or 𝐹 (c.f. Meyberg & Vachenauer, 2001; Cuyt et al., 2008; Beals & 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. 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