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LG&ItxGKl{H)>I%c diff --git a/wtheo7.tex b/wtheo7.tex index 1097653..321ec07 100644 --- a/wtheo7.tex +++ b/wtheo7.tex @@ -62,22 +62,35 @@ Daher gilt $AX + b \sim N_{A\mu + b, A\Sigma A^t}$. \item Wir definieren $E_{ij}$ mit $(E_{ij})_{kl} = \delta_{ik}\delta_{jl}$. Dann erhalten wir durch $P_{ij} = \sum_{i = 1}^{n} E_{ii} - E_{ii} - E_{jj} + E_{ij} + E_{ji}$ eine Permutationsmatrix. - Für $Y = P_{1i}X$ gilt $Y \sim N_{(P_{1i}\mu, P_{1i}\Sigma P_{i1})} = N_{(P_{1i}\mu, \Sigma)}$. - Da $\Sigma = LDL^t$ für eine normiert untere Dreiecksmatrix $L$ und eine Diagonalmatrix $D$, erhalten wir + Für $Y = P_{1i}X$ gilt $Y \sim N_{(P_{1i}\mu, P_{1i}\Sigma P_{i1})}$. + Wegen $P_{1i}\Sigma P_{i1} = LDL^t$ für eine normierte untere Dreiecksmatrix $L$ und eine Diagonalmatrix $D$, erhalten wir \[ - L^{-1}Y - L^{-1}P_i\mu \sim N_{(L^{-1} \mu - L^{-1}\mu, L^{-1}L D L^t L^{-t})} = N_{(0, D)} + L^{-1}P_{1i}X - L^{-1}P_{1i}\mu \sim N_{(L^{-1} P_{1i}\mu - L^{-1}P_{1i}\mu, L^{-1}L D L^t L^{-t})} = N_{(0, D)} \] - Es gilt + Da $L$ und somit auch $L^{-1}$ normierte untere Dreiecksmatrizen sind, gilt $(L^{-1}\cdot A)_{11} = A_{11}$ für beliebiges $A$. Daher erhalten wir für die Randverteilung \begin{align*} - \mathbbm{f}^{(L^{-1}Y - L^{-1}P_i\mu)_1}(x_1) &= \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \mathbbm{f}^{L^{-1}Y - L^{-1}P_i\mu}(x_1, \dots, x_n) \d{x_n} \d{x_2}\\ - &= \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot (\det D)^{-\frac{1}{2}} e^{-\frac{1}{2}\langle D^{-1}x, x\rangle} \d{x_n} \cdots \d{x_2}\\ - &= (\det D)^{-\frac{1}{2}}\prod_{i = 2}^{n} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}\sum_{i = 2}^{n} D_{ii}^{-1}x_i^2}\\ + \mathbbm{f}^{(L^{-1}P_{1i}X - L^{-1}P_{1i}\mu)_1}(x_1) &= \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \mathbbm{f}^{L^{-1}P_{1i}X - L^{-1}P_{1i}\mu}(x_1, \dots, x_n) \d{x_n} \d{x_2}\\ + \mathbbm{f}^{(P_{1i}X)_1 - (P_{1i}\mu)_1}(x_1)&= \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot (\det D)^{-\frac{1}{2}} e^{-\frac{1}{2}\langle D^{-1}x, x\rangle} \d{x_n} \cdots \d{x_2}\\ + \mathbbm{f}^{X_i - \mu_i}(x_1)&= (\det D)^{-\frac{1}{2}} \cdot \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}D_{11}^{-1}x_1^2}\prod_{i = 2}^{n} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}D_{ii}^{-1}x_i^2}\\ &= \frac{\sqrt{D_{22}\cdot \dots \cdot D_{nn}}}{\sqrt{\det D}} \cdot \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}D_{11}^{-1} x_1^2}\\ &= \frac{1}{\sqrt{2\pi D_{11}}}e^{-\frac{1}{2D_{11}} x_1^2}\\ - \mathbbm{f}^{X_i - \mu_i}(x_i)&\sim N_{(0, D_{11})} = N_{(0, Y_{11})} = N_{(0, X_{ii})}\\ - \mathbbm{f}^{X_i} &\sim N_{(\mu_i, X_{ii})} + \implies \mathbbm{f}^{X_i - \mu_i}(x_i)&\sim N_{(0, D_{11})} = N_{(0, (P_{1i}\Sigma P_{i1})_{11})} = N_{(0, \Sigma_{ii})}\\ + \mathbbm{f}^{X_i}(x_i) &\sim N_{(\mu_i, \Sigma_{ii})} \end{align*} - \item c + \item Durch die Permutation $P_{1i}P_{2j}$ können wir (analog zur Aufgabe (b)) o.B.d.A. $i= 1$, $j = 2$ annehmen. Wir wenden erneut die Cholesky-Zerlegung an und erhalten $\Sigma = LDL^t$, wobei wegen $\Sigma_{12} = \Sigma_{21}$ auch $L_{21} = 0$ gelten muss. + Daher ist auch $D_{11} = \Sigma_{11}$ und $D_{22} = \Sigma_{22}$. + Wir berechnen nun die Randverteilung für $(X_1, X_2)$. + \begin{align*} + \mathbbm{f}^{(L^{-1}X - L^{-1}\mu)_{(1,2)}}(x_1, x_2) &= \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \mathbbm{f}^{L^{-1}X - L^{-1}\mu}(x_1, \dots, x_n) \d{x_n} \d{x_3} + \intertext{Nun nutzen wir $L_{21} = 0$} + \mathbbm{f}^{(X_1, X_2) - (\mu_1, \mu_2)}(x_1, x_2)&= \int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot (\det D)^{-\frac{1}{2}} e^{-\frac{1}{2}\langle D^{-1}x, x\rangle} \d{x_n} \cdots \d{x_3}\\ + \mathbbm{f}^{(X_1 - \mu_1, X_2 - \mu_2)}(x_1, x_2)&= (\det D)^{-\frac{1}{2}} \cdot \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}D_{11}^{-1}x_1^2}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}D_{22}^{-1}x_2^2}\cdot \prod_{i = 3}^{n} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}D_{ii}^{-1}x_i^2}\\ + &= \frac{\sqrt{D_{33}\cdot \dots \cdot D_{nn}}}{\sqrt{\det D}} \cdot \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}D_{11}^{-1} x_1^2} \cdot \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}D_{22}^{-1} x_2^2}\\ + &= \frac{1}{\sqrt{2\pi D_{11}}}e^{-\frac{1}{2D_{11}} x_1^2}\cdot \frac{1}{\sqrt{2\pi D_{22}}}e^{-\frac{1}{2D_{22}} x_2^2}\\ + \mathbbm{f}^{(X_1, X_2)}(x_1, x_2)&= \frac{1}{\sqrt{2\pi \Sigma_{11}}}e^{-\frac{1}{2\Sigma_{11}} (x_1 - \mu_1)^2}\cdot \frac{1}{\sqrt{2\pi \Sigma_{22}}}e^{-\frac{1}{2\Sigma_{22}} (x_2 - \mu_2)^2}\\ + &= \mathbbm{f}^{X_1}(x_1) \cdot \mathbbm{f}^{X_2}(x_2) + \end{align*} + Also sind $X_1$ und $X_2$ unabhängig. \end{enumerate} \end{aufgabe} \end{document}