Formelsammlung Mathematik: Wahrscheinlichkeitsverteilungen

Diskrete Verteilungen

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

Stetige Verteilungen

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