Content Tags

There are no tags.

On the Conditional Distribution of a Multivariate Normal given a Transformation - the Linear Case.

RSS Source
Authors
Rajeshwari Majumdar, Suman Majumdar

We show that the orthogonal projection operator onto the range of the adjointof a linear operator $T$ can be represented as $UT,$ where $U$ is an invertiblelinear operator. Using this representation we obtain a decomposition of aNormal random vector $Y$ as the sum of a linear transformation of $Y$ that isindependent of $TY$ and an affine transformation of $TY$. We then use thisdecomposition to prove that the conditional distribution of a Normal randomvector $Y$ given a linear transformation $\mathcal{T}Y$ is again a multivariateNormal distribution. This result is equivalent to the well-known result thatgiven a $k$-dimensional component of a $n$-dimensional Normal random vector,where $k

Stay in the loop.

Subscribe to our newsletter for a weekly update on the latest podcast, news, events, and jobs postings.