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19 lines
662 B
TeX
19 lines
662 B
TeX
\subsection{Diskrete Zufallsvariablen}
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\shorttheorem Für $\cX$ mit Werten fast sicher in $W$:
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\[
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\E[\cX] = \sum_{x \in W} x \cdot \P[\cX = x] = \sum_{x \in W} x \cdot p_\cX(x)
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\]
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\shortremark $\E[\cX]$ wohldefiniert falls $(x \cdot p_\cX(x))_{x \in W}$ abs. konv.
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\subsubsection{Beispiele}
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\begin{itemize}
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\item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$
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\item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$
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\item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$
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\end{itemize}
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\subsubsection{Transformierte Zufallsvariablen}
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\shorttheorem Für $\varphi : \R \rightarrow \R$, $\E[\varphi(\cX)] = \sum_{x \in W} \varphi(x) \cdot \P[\cX = x]$
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