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\[E(\mathbf{X}^{T}\mathbf{A}\mathbf{X})=\textrm{trace}(\mathbf{A}\boldsymb…
\[E(\mathbf{X}^{T}\mathbf{A}\mathbf{X})=\textrm{trace}(\mathbf{A}\boldsymbol{\Sigma})+\boldsymbol{\mu}^{T}\mathbf{A}\boldsymbol{\mu}\]
Bevis
Eftersom #\[E(X_{i}X_{j})=\sigma_{ij}+\mu_{i}\mu_{j}\]så har vi\[E\left(\sum_{i=1}^{n}\sum_{j=1}^{n}X_{i}X_{j}a_{ij}\right)=\sum_{i=1}^{n}\sum_{j=1}^{n}\sigma_{ij}a_{ij}+\sum_{i=1}^{n}\sum_{j=1}^{n}\mu_{i}\mu_{j}a_{ij}\]\[=\textrm{trace}(\mathbf{A}\boldsymbol{\Sigma})+\boldsymbol{\mu}^{T}\mathbf{A}\boldsymbol{\mu}\]
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\[\left[\begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array}\right]\left[\begin{array}{cc} \sigma_{11} & \sigma_{12}\\ \sigma_{21} & \sigma_{22} \end{array}\right]=\left[\begin{array}{cc} {\color{green}{a_{11}\sigma_{11}+a_{12}\sigma_{\underbrace{21}_{=12}}}} & a_{11}\sigma_{12}+a_{12}\sigma_{22}\\ a_{21}\sigma_{11}+a_{22}\sigma_{21} & {\color{green}{a_{21}\sigma_{\underbrace{12}_{21}}+a_{22}\sigma_{22}}} \end{array}\right]\]
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