Closed form expression of the multivariate standard Normal distribution under a weighted sum constraint

2018/01/19 by Vrins, Frédéric
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1801.06387

Abstract

In this letter we derive the (n-1)-dimensional distribution corresponding to a n-dimensional i.i.d. Normal standard vector Z=(Z₁,Z₂,…,Z_n) subjected to the weighted sum constraint ∑_i=1ⁿ w_i Z_i=c, w_i≠ 0. We first address the n=2 case before proceeding with the general n≥ 2 case. The resulting distribution is a Normal distribution whose mean vector μ and covariance matrix Σ are explicitly derived as a function of w₁,…,w_n,c. The derivation of the density relies on a very specific positive definite matrix for which the determinant and inverse can be computed analytically.

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