Diskrete Verteilungen

Name	Zähldichte	Erwartungswert	Varianz
Binomial-Verteilung	${\displaystyle P(X=k)={\binom {n}{k}}\cdot p^{k}\cdot (1-p)^{n-k}}$	${\displaystyle n\cdot p}$	${\displaystyle n\cdot p\cdot (1-p)}$
Geometrische Verteilung[1]	${\displaystyle P(X=k)=p\cdot (1-p)^{k-1}}$	${\displaystyle {\frac {1}{p}}}$	${\displaystyle {\frac {1-p}{p^{2}}}}$
Hypergeometrische Verteilung	${\displaystyle P(X=k)={\frac {{\binom {M}{k}}\cdot {\binom {N-M}{n-k}}}{\binom {N}{n}}}}$	${\displaystyle n\cdot {\frac {M}{N}}}$	${\displaystyle n\cdot {\frac {M}{N}}\cdot \left(1-{\frac {M}{N}}\right)\cdot {\frac {N-n}{N-1}}}$
Negative Binomial-Verteilung[2]	${\displaystyle P(X=k)={\binom {k-1}{r-1}}\cdot p^{r}\cdot (1-p)^{k-r}}$	${\displaystyle {\frac {r}{p}}}$	${\displaystyle {\frac {r\cdot (1-p)}{p^{2}}}}$
Poisson-Verteilung	${\displaystyle P(X=k)={\frac {\lambda ^{k}}{k!}}\cdot e^{-\lambda }}$	${\displaystyle \lambda }$	${\displaystyle \lambda }$

Quellen

1. ↑ Es existieren zwei verschiedene Definitionen der geometrischen Verteilung, siehe Wikipedia
2. ↑ Auch zu der negativen Binomial-Verteilung existiert eine alternative Definition, siehe Wikipedia

Stetige Verteilungen

Name	Dichte	Erwartungswert	Varianz
Erlang-Verteilung	${\displaystyle f(x)={\frac {\lambda ^{n}\cdot x^{n-1}}{(n-1)!}}\cdot e^{-\lambda \cdot x},x\geq 0}$	${\displaystyle {\frac {n}{\lambda }}}$	${\displaystyle {\frac {n}{\lambda }}^{2}}$
Exponential-Verteilung: ${\displaystyle {\mathcal {E}}(\lambda )}$	${\displaystyle f(x)=\lambda \cdot e^{-\lambda \cdot x},x\geq 0}$	${\displaystyle {\frac {1}{\lambda }}}$	${\displaystyle {\frac {1}{\lambda ^{2}}}}$
Gamma-Verteilung: ${\displaystyle \Gamma (\alpha ,\beta )}$	${\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\cdot x^{\alpha -1}\cdot e^{-\beta \cdot x},x>0}$	${\displaystyle {\frac {\alpha }{\beta }}}$	${\displaystyle {\frac {\alpha }{\beta ^{2}}}}$
Gleichverteilung: ${\displaystyle {\mathcal {U}}(a,b)}$	${\displaystyle f(x)={\frac {1}{b-a}},x\in [a,b]}$	${\displaystyle {\frac {a+b}{2}}}$	${\displaystyle {\frac {1}{12}}\cdot (b-a)^{2}}$
Normal-Verteilung: ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$	${\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\cdot e^{-{\frac {(x-\mu )^{2}}{2\cdot \sigma ^{2}}}}}$	${\displaystyle \mu }$	${\displaystyle \sigma ^{2}}$
Lognormal-Verteilung: ${\displaystyle {\mathcal {LN}}(\mu ,\sigma ^{2})}$	${\displaystyle f(x)={\frac {1}{{\sqrt {2\pi \sigma ^{2}}}\cdot x}}\cdot e^{-{\frac {(\ln(x)-\mu )^{2}}{2\cdot \sigma ^{2}}}}}$	${\displaystyle e^{\mu +{\frac {\sigma ^{2}}{2}}}}$	${\displaystyle e^{2\mu +\sigma ^{2}}\cdot (e^{\sigma ^{2}}-1)}$