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Here, we provide a model that is called the simple linear regression model. Our model is \begin{align} Y_i = \beta_0+\beta_1 x_i +\epsilon_i, \end{align} where we model $\epsilon_i$'s as independent and zero-mean normal random variables, \begin{align} \epsilon_i \sim N(0,\sigma^2). \end{align} The parameters $\beta_0$, $\beta_1$, and $\sigma^2$ are considered fixed but unknown. The assumption is that we have data points $(x_1,y_1)$, $(x_2,y_2)$, $\cdots$, $(x_n,y_n)$ and our goal is to find the "best" values for $\beta_0$ and $\beta_1$ resulting in the line that provides the "best" fit for the data points. Here, $y_i$'s are the observed values of the random variables $Y_i$'s. To have a well-defined problem we add the following assumptions. We assume $n \geq 3$. We also assume that not all $x_i$'s are identical.

There are several common methods for finding good values for $\beta_0$ and $\beta_1$. These methods will result in the same answers; however, they are philosophically based on different ideas. Here, we will provide two methods for estimating $\beta_0$ and $\beta_1$. A third method will be discussed in the Solved Problems section.