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$\mathcal{N}$ +\begin{center} + \includegraphics[width=0.5\linewidth]{res/normal-distribution.png} +\end{center} + +\lemma \textbf{Summe von Normalverteilten ist normalverteilt}\\ +\smalltext{$X_1,\ldots,X_n$ unabh. s.d. $X_i \sim \mathcal{N}(m_i,\sigma_i^2)$} +\begin{align*} + Z &= m_0 + \lambda_1X_1 + \ldots + \lambda_nX_n \\ + Z &\sim \mathcal{N}\Biggl( \underbrace{m_0 + \sum_{i=1}^n \lambda_iX_i}_{m_Z}\quad,\quad \underbrace{\sum_{i=1}^n \lambda_i^2 \sigma_i^2}_{\sigma^2_Z} \Biggr) +\end{align*} + +\definition \textbf{Standardnormalverteilung} +$$X \sim \mathcal{N}(0,1) \qquad \Phi(x) := \P[X \leq x]$$ + +\lemma \textbf{Symmetrie} $\quad \Phi(-x) = 1-\Phi(x)$ + + +\lemma \textbf{Normalverteilung via } $\mathcal{N}(0,1)$\\ +\smalltext{$Z \sim \mathcal{N}(m, \sigma^2) \text{ arbiträr},\quad X \sim \mathcal{N}(0,1)$} +$$ + Z = m + \sigma\cdot X +$$ + +\lemma \textbf{Standardisierung}\\ +\smalltext{$Z \sim \mathcal{N}(m, \rho^2),\quad x\in\R$} +$$ + \P[x\leq Z] = \Phi\Biggl( \frac{x - m}{\rho} \Biggr) +$$ \ No newline at end of file diff --git a/semester4/ps/ps-rb/parts/03_expectation.tex b/semester4/ps/ps-rb/parts/03_expectation.tex index 5f13b12..59eed22 100644 --- a/semester4/ps/ps-rb/parts/03_expectation.tex +++ b/semester4/ps/ps-rb/parts/03_expectation.tex @@ -128,6 +128,15 @@ $$ \mathbb{V}[X] = \E[X^2]-\E[X]^2 $$ +{\footnotesize + \textbf{Beispiel:} Bestimme $\E\bigl[X^2\bigr]$, wobei $X \sim \text{Poisson}(\lambda)$, $\lambda > 0$. + \begin{align*} + \V[X] &= \E\bigl[X^2\bigr] - \E[X]^2 & (X \sim \text{Poisson}(\lambda)) \\ + \lambda &= \E\bigl[X^2\bigr] - \lambda^2 \\ + \lambda + \lambda^2 &= \E\bigl[X^2\bigr] \\ + \end{align*} +} + \lemma \textbf{Eigenschaften} \begin{align*} \text{(i)} &\quad \V[X] \geq 0 \\ diff --git a/semester4/ps/ps-rb/res/normal-distribution.png b/semester4/ps/ps-rb/res/normal-distribution.png new file mode 100644 index 0000000000000000000000000000000000000000..d7fd4fb216af6860888bf295c5c59b43a0b7cd6b GIT binary patch literal 12847 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