Aufgabe:
Gegeben ist \( x=\left(\begin{array}{l}{X_{1}} \\ {X_{2}} \\ {X_{3}}\end{array}\right), \operatorname{Var}(\boldsymbol{x})=\boldsymbol{\Sigma}=\left(\begin{array}{ccc}{\sigma_{1}^{2}} & {\sigma_{12}} & {\sigma_{13}} \\ {\sigma_{21}} & {\sigma_{2}^{2}} & {\sigma_{23}} \\ {\sigma_{31}} & {\sigma_{32}} & {\sigma_{3}^{2}}\end{array}\right) \)
Problem/Ansatz:
Berechnen Sie die Varianzen von X1+X2.
Musterlölung:
\( \operatorname{Var}\left(X_{1} \pm X_{2}\right)=\operatorname{Var}[\left(\begin{array}{ccc}{1} & {\pm 1} & {0}\end{array}\right)\left(\begin{array}{l}{X_{1}} \\ {X_{2}} \\ {X_{3}}\end{array}\right)]=\left(\begin{array}{lll}{1} & {\pm 1} & {0}\end{array}\right)\left(\begin{array}{ccc}{\sigma_{1}^{2}} & {\sigma_{12}} & {\sigma_{13}} \\ {\sigma_{21}} & {\sigma_{2}^{2}} & {\sigma_{23}} \\ {\sigma_{31}} & {\sigma_{32}} & {\sigma_{3}^{2}}\end{array}\right)\left(\begin{array}{c}{1} \\ {\pm 1} \\ {0}\end{array}\right) \)
Warum (1 1 0)und(1 1 0)-1?
