Introduction

In statistics and machine learning, understanding the relationships between variables is crucial for building predictive models and analyzing data. One of the basic techniques for exploring these relationships is the bivariate projection, which relies on the concept of the bivariate normal distribution. This technique allows for the examination and prediction of the behavior of one variable in terms of another, utilizing the dependency structure between them.

Bivariate projection helps determining the expected value of one random variable given a specific value of another variable. For instance, in linear regression, projection helps estimate how a dependent variable changes with respect to an independent variable.

This article is divided into 3 parts: in the first part, I will explore the fundamentals of bivariate projection, deriving its formulation and demonstrating its application in regression models. In the second part, I will provide some intuition behind the projection and some plots to better understand its implications. In the third part, I will use the projection to derive the parameters for a linear regression.

In my derivation of the bivariate projection formula, I will use some well known results. In order not to be too heavy on the reader, I will provide the proofs…