User leo - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:29:37Z http://mathoverflow.net/feeds/user/27193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109471/expectation-of-a-multivariate-gaussian-over-a-plane Expectation of a multivariate Gaussian over a plane Leo 2012-10-12T16:57:30Z 2012-10-12T17:15:33Z <p>For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation :</p> <p>$E[X|X^Tb = c]$</p> <p>That is the expected value of the multivariate variable $X$ given that it will lie on the plane $X^Tb = c$. I have tried by parametrizing $X$ as $X = \vec{a_0} + t_1 \vec{a_1} ... t_{n-1} \vec{a_{n-1}}$ and calculating the integral $\int_{-\infty}^{\infty} ... \int_{-\infty}^{\infty} xf(x) dt_1 ... dt_{n-1}$, where $f(x)$ is the pdf of the Gaussian, but I end up with an extremely messy formula even when trying to solve in the simple three-dimensional case. </p> <p>My question is whether there is a known closed form solution for the above expectation and/or if there is a specific parametrization I could use to simplify the solution.</p>