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

TL;DR

This paper derives a closed-form expression for the distribution of a multivariate standard normal vector constrained by a weighted sum, providing explicit formulas for the mean and covariance in terms of the weights and sum constant.

Contribution

It presents the first explicit derivation of the constrained distribution's mean and covariance for any dimension, extending previous work limited to specific cases.

Findings

Explicit formulas for mean and covariance derived

Distribution is a multivariate normal with these parameters

Applicable for any dimension n ≥ 2

Abstract

In this letter we derive the $(n-1)$-dimensional distribution corresponding to a $n$-dimensional i.i.d. Normal standard vector $Z=(Z_1,Z_2,\ldots,Z_n)$ subjected to the weighted sum constraint $\sum_{i=1}^n w_i Z_i=c$, $w_i\neq 0$. We first address the $n=2$ case before proceeding with the general $n\geq 2$ case. The resulting distribution is a Normal distribution whose mean vector $\mu$ and covariance matrix $\Sigma$ are explicitly derived as a function of $w_1,\ldots,w_n,c$. The derivation of the density relies on a very specific positive definite matrix for which the determinant and inverse can be computed analytically.