On the Conditional Expectation of Mean Shifted Gaussian Distributions

TL;DR

This paper investigates how the conditional expectation of Gaussian distributions shifts when the mean is moved, demonstrating that the expectation shifts correspondingly under the same conditioning event.

Contribution

It introduces a property of Gaussian distributions showing the direct relationship between mean shifts and conditional expectation shifts under identical conditions.

Findings

Conditional expectation shifts to the right with mean shifts to the right.

The property holds for univariate Gaussian distributions under the same conditioning.

Provides theoretical insight into Gaussian distribution behavior under mean shifts.

Abstract

In this paper, we consider a property of univariate Gaussian distributions namely conditional expectation shift (or centroid shift). Specifically, we compare two Gaussian distributions in which they differ only in their means. Equivalently, we can view this situation as one of the distribution is shifted to the right. These two distributions are conditioned on the same event in which the realizations fall in the right interval or left interval. We show that if a Gaussian distribution is shifted to the right while the conditioning event remains the same then the conditional expectation is shifted to the right concurrently.