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.

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