# Theoretical regression line - how does it look like graphically

On the image there is the empirical regression line (the best fit by sum of min squares - $y_i - \hat{y}$).

But how would the theoretical regression line look like? Does it join all the points? I guess, then it would not be line.

Thanks

Assuming the points are generated by adding i.i.d. Gaussian noise $\epsilon_i$ to the true regression line, you cannot know what the true regression line is without more information, e.g., knowing what the actual value of the Gaussian noise terms $\epsilon_i$. After all, that is the point of regression; if we knew what the true regression line were, we wouldn't need to do least squares.
Note that the real regression line is the conditional expectation, i.e., Let's say that you are observing $\{(X_i, Y_i)\}_{i=1}^n$, and you are interested to predict $Y$ using $X$, as such by the orthogonal decomposition $$Y= E[Y|X]+\epsilon.$$ Note that $E[\epsilon|X]=E[Y - E[Y|X] |X]=0$, thus by using regression you are trying to estimate the conditional mean $E[Y|X]$. If $(Y,X)$ is $MVN(\mu, \Sigma)$, then the error term is Guassian and the conditional mean is indeed a straight line $$E[Y|X] = \beta_0 + \beta_1 X,$$ where $\beta_0 = \mu_X - \beta_1 \mu_X$ and $\beta_1 = cov(X,Y)/\sigma^2_X$. For non Gaussian setting, the conditional mean can be non-linear, thus the regression $$Y = \hat{\beta}_0 + \hat{\beta}_1X,$$ is only a linear approximation of $E[Y|X]=g(X)$ which quality depends on the structure of $g$.