I have a set of $N$ points $(x_i,y_i)$. $X$ and $Y$ both have some noise associated with them due to measurement inaccuracy however the relationship of the underlying true values (i.e. if we could remove the noises) of these points should be of the form $y = mx +c$ where $m$ and $c$ are constants.
However, due to measurement inaccuracies in both $X$ and $Y$, I will get uncertainties in both my $m$ and $c$ values.
1)If I assume that my measurement inaccuracies for both $X$ and $Y$ are Gaussian distributed $\epsilon$ ~ $N(0,\sigma)$ how do I obtain the most likely $(m,c)$ and the uncertainties/confidence in both.
2) If I instead know that the uncertainties are different for $X$ and $Y$ such that $\sigma_x \neq \sigma_y$ where $\epsilon_x$ ~ $N(0,\sigma_x)$, $\epsilon_y$ ~ $N(0,\sigma_y)$ can I get a different estimate of $(m,c)$ and the uncertainties/confidence in both.