preprint · arXiv (Cornell University) · 2016

On the regular conditional distribution of a multivariate Normal given a linear transformation

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Abstract

We show that the orthogonal projection operator onto the range of the adjoint of a linear operator T can be represented as UT, where U is an invertible linear operator. Using this representation we obtain a decomposition of a multivariate Normal random variable Y as the sum of a linear transformation of Y that is independent of TY and an affine transformation of TY. We then use this decomposition to prove that the regular conditional distribution of a multivariate Normal random variable Y given a linear transformation TY is again a multivariate Normal distribution. This result is equivalent to the well-known result that given a k-dimensional component of a n-dimensional multivariate Normal random variable, where k < n, the regular conditional distribution of the remaining (n - k)-dimensional component is a (n - k)-dimensional multivariate Normal distribution.

Cite this paper

APA 7

Majumdar, R., & Majumdar, S. (2016). On the regular conditional distribution of a multivariate Normal given a linear transformation. arXiv (Cornell University). https://doi.org/10.48550/arxiv.1612.01210

MLA 9

Majumdar, Rajeshwari, and Suman Majumdar. “On the regular conditional distribution of a multivariate Normal given a linear transformation.” arXiv (Cornell University), 2016, https://doi.org/10.48550/arxiv.1612.01210.

Chicago

Majumdar, Rajeshwari, and Suman Majumdar. 2016. “On the regular conditional distribution of a multivariate Normal given a linear transformation.” arXiv (Cornell University). https://doi.org/10.48550/arxiv.1612.01210.

Harvard

Majumdar, R. and Majumdar, S. (2016) ‘On the regular conditional distribution of a multivariate Normal given a linear transformation’, arXiv (Cornell University). Available at: https://doi.org/10.48550/arxiv.1612.01210.

BibTeX

@misc{majumdar2016regular,
  title = {{On the regular conditional distribution of a multivariate Normal given a linear transformation}},
  author = {Majumdar, Rajeshwari and Majumdar, Suman},
  journal = {arXiv (Cornell University)},
  year = {2016},
  doi = {10.48550/arxiv.1612.01210}
}

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