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I am given the equation $y = mx + c + z$ where $z \sim Normal(0,\sigma^2)$ is our error term of the linear regression.

I am given a sample with just one point $(x_1, y_1)$, and I've been asked to find the likelihood of this sample.

Additionally, I am also asked to find the likelihood of another sample where there are $n$ of $(x_i, y_i)$ points. I've learned in the past in econometrics about how to find the maximum likelihood estimators of $\beta_1$ and $\beta_0$ in $y = \beta_1x + \beta_0$, but I am struggling to see how I should approach this one. If anyone could guide me on how to approach the first question, I would greatly appreciate it.

Thanks.

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You're given that y=mx+c+z, with $z\sim N(0, \sigma^2)$. You can view this as a constant plus a normal random variable. Also each observation of y should be considered as independent from one another. We expect y to follow a normal distribution with mean $E(y)=E(mx+c+z)=mx+c+E(z)=mx+c$ and variance $var(y)=var(mx+c+z)=var(z)=\sigma^2$. Now there are y's to keep in mind, the y that you observed and the $\hat y$, predicted y from your x value.

  • The likelihood for a single observation is given by $$p(y|x)=\frac 1 {\sqrt{2\pi\sigma}}\exp-\frac 1 {2\sigma^2}\left(y-(mx+c)\right)^2$$

  • The likelihood for a sample is given by $$\begin{split}p(\textbf y|\textbf x)&=\prod\frac 1 {\sqrt{2\pi\sigma}}\exp-\frac 1 {2\sigma^2}\left(y-(mx+c)\right)^2\\ &=\left(\frac 1 {\sqrt{2\pi\sigma}}\right)^n\exp-\frac 1 {2\sigma^2}\sum_{i=1}^n\left(y_i-(mx_i+c)\right)^2\end{split}$$

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