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34 lines
1.7 KiB
TeX
34 lines
1.7 KiB
TeX
\subsection{Varianz}
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$\E[\cX^2] = \sum_{x \in W} x^2 \cdot p_\cX(x)$ ($\cX$ diskret)
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\shortdefinition $\cX$ mit $\E[\cX^2] < \8$, $\V[\cX] = \E[(\cX - \E[\cX])^2]$
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\shortdefinition[Standardabweichung] $\sigma(\cX) = \sqrt{\V[\cX]}$
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\shortremark $\V[\cX] = \E[\cX^2] - \E[\cX]^2$ ($\V[\cX] \geq 0$ ist wahr)
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\shortexample $\cX$ determ. Z.V (= konst) mit Wert $a$, also $\cX = a1_\Omega$. Dann: $\E[\cX] = a \E[1_\Omega] = a \P[\Omega] = a$ und\\
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$\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$
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% Task 4.42 (needs proof?)
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\shortremark $\E[\cX] < \8$, dann $\V[\cX] \geq 0$ mit $=$ g.d.w. $\cX$ konst; \hl{zudem} $\V[a \cX] = a^2 \V[\cX]$ und $\V[\cX + a] = \V[\cX]$
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\shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$.
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Falls $\cX_k$ abhängig, dann gilt $\neq$.
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Falls \bi{mermals} selbe Z.V. (e.g. $\cX - \cY - \cY$, ist $\cX - 2\cY$, dann $\V = 2^2 \V[\cY]$)
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\newpage
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\shortexample Varianz von bekannten Verteilungen
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\begin{itemize}
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\item $\cX \sim \text{Ber}(p)$, $\V[\cX] = p (1 - p)$
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\item $\cX \sim \text{Bin}(n, p)$, $\V[\cX] = n p (1 - p)$
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\item $\cX \sim \text{NBin}(r, p)$, $\V[\cX] = \frac{r(1 - p)}{p^2}$
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\item $\cX \sim \text{Geom}(p)$, $\V[\cX] = \frac{1 - p}{p^2}$
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\item $\cX \sim \text{H}(n, r, m)$, $\V[\cX] = m\frac{rn - r^2}{n^2} \frac{n - m}{n - 1}$
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\item $\cX \sim \text{Poisson}(\lambda)$, $\V[\cX] = \lambda = \E[\cX]$
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\item $\cX \sim \cU([a, b])$, $\V[\cX] = \frac{(b - a)^2}{12}$
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\item $\cX \sim \text{Exp}(\lambda)$, $\V[\cX] = \frac{1}{\lambda^2}$
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\item $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$
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\item $\cX \sim \text{Cauchy}(x_0, \gamma)$: Existiert nicht
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\end{itemize}
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