Benutzer:Panzi/Statistik- und Wahrscheinlichkeitstheorie

Verteilungen

Normalverteilung $N(\mu ,\sigma ^{2})$

$f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}$
$F(x)={\frac {1}{\sigma \cdot {\sqrt {2\pi }}}}\cdot \int _{-\infty }^{x}e^{-{\frac {1}{2}}\cdot \left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t$

Binomialverteilung $Bi(n,p)$

$f(x)={n \choose k}p^{k}(1-p)^{n-k}$
$F(x)=\sum _{k=0}^{k<t}{n \choose k}p^{k}(1-p)^{n-k}$

Bernoulli-Verteilung $Bi(1,p)$

$f(x)=p^{x}\cdot q^{1-x}$
$F(x)=1-p$

Poissonverteilung $P(\lambda )$

$f(x)={\frac {\lambda ^{x}}{x!}}\;{\rm {e}}^{-\lambda }$
$F(x)=e^{-\lambda }\sum _{k=0}^{x}{\frac {\lambda ^{k}}{k!}}$

Rechteckverteilung $R(a,b)$

$f(x)={\begin{cases}0&x\leq a\\{\frac {1}{b-a}}&a<x<b\\0&x\geq b\end{cases}}$
$F(x)={\begin{cases}0&x\leq a\\{\frac {x-a}{b-a}}&a<x<b\\1&x\geq b\end{cases}}$

Exponentialverteilung $Exp(\lambda )$

$f(x)={\begin{cases}\displaystyle \lambda {\rm {e}}^{-\lambda x}&x\geq 0\\0&x<0\end{cases}}$
$F(x)={\begin{cases}\displaystyle 1-\mathrm {e} ^{-\lambda x}&x\geq 0\\0&x<0\end{cases}}$

Chi-Quadrat-Verteilung $\chi ^{2}(n,\lambda )$

$f(x)={\begin{cases}\displaystyle {\frac {x^{{\frac {n}{2}}-1}e^{-{\frac {x}{2}}}}{2^{\frac {n}{2}}\Gamma \left({\frac {n}{2}}\right)}}&x>0\\0&x\leq 0\end{cases}}$
$F(x)=\Gamma \left({\frac {n}{2}},{\frac {x}{2}}\right)$

Andere Formeln

Wahrscheinlichkeitsnetz

Bei $n\leq 30$:
$Punkt=\left(x_{i},{\frac {i-0.5}{n}}\right)$

Stichprobenvarianz

$s^{2}={\frac {1}{n-1}}\sum \limits _{i=1}^{n}\left(x_{i}-{\overline {x}}\right)^{2}$

Erwartungswert bei stetigen Zufallsvariablen

$E[h(X)]=\int _{-\infty }^{\infty }h(x)f(x)dx$
$E[ah_{1}(X)+bh_{2}(X)]=E[ah_{1}(X)]+E[bh_{2}(X)]$

Varianz bei stetigen Zufallsvariablen

$\sigma ^{2}=Var(X)=E(X-\mu )^{2}=E[X-E(X)]^{2}$
bzw.
$Var(X)=E[X)^{2}-(EX)^{2}$

Standardisieren für die Z-Verteilung

$z={\frac {{\overline {x}}-\mu _{0}}{\frac {\sigma }{\sqrt {n}}}}$

Konfidenzintervalle

$\sigma ^{2}=\left({\frac {(n-1)\cdot s^{2}}{\chi _{\frac {\alpha }{2}}^{2}}},{\frac {(n-1)\cdot s^{2}}{\chi _{1-{\frac {\alpha }{2}}}^{2}}}\right)$

µ bei Z-Verteilung

$\mu ={\overline {x}}\pm z_{1-{\frac {\alpha }{2}}}{\frac {\sigma }{\sqrt {n}}}$

$n=\left(2\cdot z_{1-{\frac {\alpha }{2}}}{\frac {\sigma }{d}}\right)^{2}$

µ bei t-Verteilung

$\mu ={\overline {x}}\pm {\frac {t_{n-1,1-{\frac {\alpha }{2}}}\cdot s}{\sqrt {n}}}$

Anteilschätzung

$Anteil={\overline {x}}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\overline {x}}{\frac {1-{\overline {x}}}{n}}}}$
$Z={\frac {{\overline {X}}-p_{0}}{\sqrt {\frac {p_{0}(1-p_{0})}{n}}}}$

Tests

$Z={\frac {{\overline {X}}-\mu }{\frac {\sigma }{\sqrt {n}}}}$
$T={\frac {{\overline {X}}-\mu }{\frac {S}{\sqrt {n}}}}$

Gleichheit der Mittel zweier Stichproben

$T={\sqrt {\frac {n_{x}n_{y}(n_{x}+n_{y}-2)}{n_{x}+n_{y}}}}\cdot {\frac {{\overline {X}}-{\overline {Y}}}{\sqrt {(n_{x}-1)S_{x}^{2}+(n_{y}-1)S_{y}^{2}}}}$
$-t_{n_{x}+n_{y}-2,1-\alpha }$ Aufpassen auf das Intervall [-t,t].

Varianzanalyse

k ... Anzahl der Gruppen
$n_{i}$ ... Größe der i-ten Gruppe

$q_{I}=\sum \limits _{j=1}^{k}\sum \limits _{i=1}^{n_{j}}(x_{ij}-{\overline {x}}_{j})^{2}$
$q_{Z}=\sum \limits _{j=1}^{k}n_{j}({\overline {x}}_{j}-{\overline {x}})^{2}$
$S_{I}^{2}={\frac {q_{I}}{n-k}}$
$S_{Z}^{2}={\frac {q_{Z}}{k-1}}$
$F={\frac {S_{Z}^{2}}{S_{I}^{2}}}$
$F>F_{k-1,n-k;1-\alpha }$

Regression und Korellation

$s_{xy}={\frac {1}{n-1}}\sum \limits _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})$
${\hat {a}}={\overline {y}}$
${\hat {b}}={\frac {s_{xy}}{s_{x}^{2}}}={\frac {\sum \limits _{i=1}^{n}(x_{i}-{\overline {x}})(y_{i}-{\overline {y}})}{\sum \limits _{i=1}^{n}(x_{i}-{\overline {x}})^{2}}}$
${\hat {y}}_{x}={\hat {a}}+{\hat {b}}(x-{\overline {x}})$

$\sigma _{XY}=E[(X-\mu _{X})(Y-\mu _{Y})]$
$\rho _{XY}={\frac {\sigma _{XY}}{\sigma _{X}\sigma _{Y}}}$
$r_{XY}={\frac {s_{XY}}{s_{X}s_{Y}}}$

Test auf Unkorreliertheit

$T=R{\sqrt {\frac {n-2}{1-R^{2}}}}$
$|T|>t_{n-2;1-{\frac {\alpha }{2}}}$

Umformung um R auszurechnen:

$R={\sqrt {\frac {T^{2}}{T^{2}+n-2}}}$