Distribution of multivariate normal X given AX=b

11.08.2015, 23:24	FCunderC	Auf diesen Beitrag antworten »
Distribution of multivariate normal X given AX=b Meine Frage: Hello, there is the following problem: I have a multivariate normally distributed vector $\begin{eqnarray} X\sim(\mu,\Sigma) \end{eqnarray}$ with $\begin{eqnarray} \mu=(\tilde{\mu},0,\dots,0)^T \end{eqnarray}$ and $\begin{eqnarray} \Sigma=diag(\sigma_1^2,\dots,\sigma_M^2) \end{eqnarray}$ , a matrix $\begin{eqnarray} A\in\mathbb R^{n\times M} \end{eqnarray}$ and a vector $\begin{eqnarray} b\in\mathbb R^n \end{eqnarray}$ . What is then the distribution of $\begin{eqnarray} X\|AX=b \end{eqnarray}$ ? I think it will not be again normally distributed like $\begin{eqnarray} X \end{eqnarray}$ itself. Especially, I am intersted in marginal distributions of $\begin{eqnarray} X\|AX=b \end{eqnarray}$ , because it would be very nice, if e.g. the expectation $\begin{eqnarray} E[X_1\|AX=b] \end{eqnarray}$ could be calculated. Maybe someone has an idea or hint? Meine Ideen: The density of $\begin{eqnarray} X\|AX=b \end{eqnarray}$ is $\begin{eqnarray} f_{X\|AX=b}(x)=\frac{f_{X,AX}(x,b)}{f_{AX}(b)} \end{eqnarray}$ . $\begin{eqnarray} AX \end{eqnarray}$ is again multivariate normal, but what is with $\begin{eqnarray} (X,AX) \end{eqnarray}$ (it is not normally distr.)?

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