# If $\epsilon_i \sim \mathcal{N}(0, \sigma^2)$, why does this also imply $x_i|\beta \sim \mathcal{N}(0, \sigma^2)$

I have seen this stated in multiple sources, where if the errors in a linear model ($$y_i = \beta x_i + \epsilon_i$$) follow $$\epsilon_i \sim \mathcal{N}(0, \sigma^2)$$, then $$x_i|\beta \sim \mathcal{N}(0, \sigma^2)$$, the same distribution. Here is one link https://www.youtube.com/watch?v=_-Gnu498s3o that states this, starting at around 2:20.

If the error terms are gaussian distributed, why does this imply that the independent and dependent variables are also gaussian distributed?

• I've watched the video and I believe the creator of the video made a typo when they wrote the likelihood. He wrote $f(x_i | \beta, \sigma^2)$ but he should have wrote $f(y_i | \beta, \sigma^2)$. That is, $$y_i | \beta, \sigma^2 \sim \mathcal N (\beta x_i , \sigma^2)$$ – SOULed_Outt May 25 '20 at 6:39
• Also, depending on how you'd like to treat $x_i$ you may want to write the likelihood as $f(y_i | x_i, \beta, \sigma^2)$. – SOULed_Outt May 25 '20 at 6:40
• @SOULed_Outt Ah, maybe that's what we meant. I think the latter form you commented is the one that I see the most often. Although, I think I usually see the semicolon usage $f(y_i | x_i ; \beta)$. Andrew NG's CS229 notes uses the semicolon notation to indicate that we're not conditioning on $\beta$. – user5965026 May 25 '20 at 6:43
• Perhaps the notation is to make it clearer that you're treating the parameters as fixed values (i.e. not random variables). Then it would be better to say $$y_i | x_i; \beta, \sigma^2 \sim \mathcal N (\beta x_i, \sigma^2)$$ and $$f(y_i | x_i; \beta, \sigma^2)$$ – SOULed_Outt May 25 '20 at 8:06

It doesn’t imply anything about the predictors (independent variables) or the response (dependent variable). It is a comment about the conditional distribution of $$y$$, conditioned on some specified value of $$x$$.
The idea is that you’re sliding a bell curve up and down the regression line. For example, The regression line gives the expected value, but then you draw an observation from the conditional distribution of $$y$$ given that $$x$$-value. That’s where the error comes from.
Remember that this framework posits that the conditional distribution is $$N(\hat{y}_i, \sigma^2)$$.
• Ah I think I get it. So basically the idea is by assuming the errors are normally distributed with zero mean, means that on average, our $y_i$ will fall on the regression line. Is this regression line determined from population parameters or from sample parameters? Also do you know why the video states $f(x_i|\beta)$ is normally distributed. I was really confused by that. – user5965026 May 25 '20 at 5:58