7.2 Mathematical Model

The mathematical model that we use for regression has three features.

  1. \(Y\) values are normally distributed at any given \(X\)
  2. The mean of \(Y\) values at any given \(X\) follows a straight line \(Y = \beta_{0} + \beta_{1} X\).
  3. The variance of \(Y\) values at any \(X\) is \(\sigma^2\) (same for all X). This is known as homoscedasticity, or homogeneity of variance.

Mathematically this is written as:

\[ Y|X \sim N(\mu_{Y|X}, \sigma^{2}) \\ \mu_{Y|X} = \beta_{0} + \beta_{1} X \\ Var(Y|X) = \sigma^{2} \]

and can be visualized as:

Figure 6.2

7.2.1 Unifying model framework

The mathematical model above describes the theoretical relationship between \(Y\) and \(X\). So in our unifying model framework to describe observed data,

DATA = MODEL + RESIDUAL

Our observed data values \(y_{i}\) can be modeled as being centered on \(\mu_{Y|X}\), with normally distributed residuals.

\[ y_{i} = \beta_{0} + \beta_{1} X + \epsilon_{i} \\ \epsilon_{i} \sim N(0, \sigma^{2}) \]