Variance of the sum of independent random variables in spheres

Abstract

The sum of random variables (errors) is the key element both for its statistical study and for the estimation and control of errors in many scientific and technical applications. In this paper we analyze the sum of independent random variables (independent errors) in spheres. This type of errors are very important, for example, in quantum computing. We prove that, given two independent isotropic random variables in an sphere, X₁ and X₂, the variance verifies $V(X_1+X_2)=V\left(X_1\right)+V\left(X_2\right)-\frac{V\left(X_1\right)V\left(X_2\right)}{2}$ and we conjecture that this formula is also true for non-isotropic random variables.