Simple linear Regression And Multiple Linear Regression

A linear regression model with one explanatory variable is referred to as a simple linear regression model.

A linear regression model with more than one explanatory variable is referred to as a multiple regression model.

Notation

Regression models can be expressed in several ways. For example, the quadratic regression model used in the potatoes example with response variable $y$ and predictor variables $x$ can be expressed as

$$y_i = \alpha + \beta x_i+ \gamma x_i^2+ \epsilon_i, \quad \quad i=1,\dots,n.$$

with various assumptions regarding $\epsilon_1, \ldots, \epsilon_n$ that we will discuss later. What we really mean to say is that given a value $x$, we expect the value of $y$ to be $\alpha + \beta x+ \gamma x^2$.

We may write this as $E(y_i |x_i)=\alpha + \beta x_i+ \gamma x_i^2$.

More generally, for a simple linear regression with response $y$ and explanatory variable $x$, we may write

$$\begin{eqnarray*} E(y_i|x_i) &=& \alpha + \beta x_i. \end{eqnarray*}$$

To ease notation, we may expression our model as $$\begin{eqnarray*} E(y_i) = \alpha + \beta x_i\\ \end{eqnarray*}$$

for $i=1, \ldots, n.$