Benutzer:Bigbossfarin/test

• VRN OSM
• Rosensteintunnel
• Studierendenverbindungen
• Liste der stillgelegten Eisenbahnstrecken in Baden-Württemberg
• Stäffele
• Commons Stuttgart, Heidelberg
• Liste Brauereien
• Untappd in Wikidata
• la:Vicipaedia:Paginae quas omnibus Wikipediis contineri oportet/Expansio/Mathematica

• erledigt für 254/423 Hochschulen, also für alle, die Koordinaten in Wikidata haben. <mapframe latitude="51.296276" longitude="10.195313" zoom="5" width="230" height="286" align="right">{

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{

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^▼49.40878.705
49.40878.705

Gemischte Poisson-Verteilung	Mischende Verteilung	Referenz
Negativ binomial	Gamma	Greenwood & Yule (1920)
Geometrisch	Exponential	Johnson et al. (1992)
Poisson-Lognormal	Lognormal	Bulmer (1974)
Poisson-Uniform	Gleich	Bhattacharya (1966)
Sichel	Invers Normal	Sichel (1975)
Neyman	Poisson	Douglas (1980)
Poisson–Pareto	Pareto	Willmot (1993)
Poisson-Shifted Pareto	Verschoben Pareto	Willmot (1993)
Poisson–Lomax	Lomax	Al-Awadhi & Ghitany (2001)
Poisson-Generalized Inverse Gaussian	Generalized Inverse Gaussian	Sichel (1974)
Poisson-Inverse Gamma	Invers Gamma	Willmot (1993)
Poisson-Truncated Normal	Truncated Normal	Patil (1964)
Poisson-Generalized Pareto	Verallgemeinert Pareto	Kempton (1975)
Poisson–Pearson Family	Pearson’s Family of Distributions	Albrecht (1982)
Delaporte	Shifted Gamma	Ruohonen (1988)
Poisson-Linear Exponential Family	Linear Exponential Family	Sankaran (1969)
Poisson–Lindley	Lindley	Sankaran (1970)
Poisson-Linear Exponential	Linear Exponential	Kling & Goovaerts (1993)
Poisson-Confluent Hypergeometric Series	Confluent Hypergeometric Series	Bhattacharya (1966)
Generalized Waring	Gamma Product Ratio	Irwin (1975)
Simple Waring	Exponential Beta	Pielou (1962)
Yule	Beta with Specific Parameter Values	Simon (1955)
Poisson-Beta I	Beta Type I	Holla & Bhattacharya (1965)
Poisson-Beta II	Beta Type II	Gurland (1958)
Poisson-Truncated Beta II	Truncated Beta Type II	Willmot (1986)
Poisson-Truncated Gamma	Truncated Gamma	Willmot (1993)
Poisson-Generalized Gamma	Generalized Gamma	Albrecht (1984)
Poisson-Modified Bessel of the 3rd Kind	Modified Bessel of the 3rd Kind	Ong & Muthaloo (1995)
Poisson-Log-Student	Log-Student	Gaver & O’Muircheartaigh (1987)
Poisson-Power Function	Power Function Distribution	Rai (1971)
Poisson-Power Variance	Power Variance Family	Hougaard et al. (1997)
	Other Discrete Distributions	Johnson et al. (1992)

[1]

Sei $X_{1},\dots ,X_{n}\sim U[a,b]$ dann ist ${\hat {a}}_{ML}=\min\{X_{1},\dots ,X_{n}\}$ und ${\hat {b}}_{ML}=\max\{X_{1},\dots ,X_{n}\}=X_{(n)}$.
Beweis:
$L_{n}(a,b)=\prod _{i=1}^{n}f(x_{i})={\frac {1}{(b-a)^{n}}}\mathbf {1} _{[a,b]}(x_{1},\dots ,x_{n})={\frac {1}{(b-a)^{n}}}\mathbf {1} _{\{a\leq \min\{x_{1},\dots ,x_{n}\}\}}\mathbf {1} _{\{\max\{x_{1},\dots ,x_{n}\}\leq b\}}$.
Da $n\geq 1$ wird der Faktor ${\frac {1}{(b-a)^{n}}}$ durch die größtmögliche Wahl von $a$ maximiert, dieses ist in $L_{n}(a,b)$ nach oben jedoch durch $\min\{x_{1},\dots ,x_{n}\}$ beschränkt. Daher liefert $a=\min\{x_{1},\dots ,x_{n}\}$ das Maximum von $L_{n}(a,b)$.
Analog folgt die Aussage für $b=\max\{x_{1},\dots ,x_{n}\}$.