Simple linear regression

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The basic problem in regression analysis is to understand the relationship between a response variable, denoted by $Y$, and one or more predictor variables, denoted by $X$. The relationship is typically empirical or statistical as opposed to functional or mathematical. The goal is to describe this relationship in the form of a functional dependence of the mean value of Y given any value of $X$ from paired observations $ \{(X_i,Y_i) : i=1,\ldots,n\} $

A basic linear regression model for the response $Y$ on the predictor $X$ is given by

$$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i, \qquad i=1,\ldots,n,$$

where the noise $ \varepsilon_1, \ldots, \varepsilon_n $ are uncorrelated, $Mean(\varepsilon_i) = 0$, and $ Variance(\varepsilon_i) = \sigma^2.$

Interpretation

Look at the scatter plot of $Y$ (vertical axis) vs. $X$ (horizontal axis). Consider narrow vertical strips around the different values of $X$:

Mean (measure of center) of the points falling in the vertical strips lie (approximately) on a straight line with slope $ \beta_1$ and intercept $\beta_0$ .
Standard deviations (measure of spread) of the points falling in each vertical strip are (roughly) the same.

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