Consider systematic error in a fit My question arise directly from a lab situation.
Let's say I have measured two sets of data $x_i$, $y_i$ and i know both the statistic and systematic error for $y_i$ (I assume no error on $x_i$). What I want to do is a fit to a function $f(x)=ax+b$ and express the result in the form of $a\pm syst \pm stat$. 
 A: Normally a linear model with statistical error is expressed as 
$$y=ax+b+\epsilon$$
Where $\epsilon$ is a random variable with a normal distribution. $\epsilon \sim N(0,\sigma^2)$
If you have an added systematic error such as a measuring instrument which adds some noise to the true value then you can model this as an additional normally distributed random variable $\tau \sim N(\mu_{sys},\sigma_{sys}^2)$
I'm assuming that your instrument is uncalibrated with $\mu_2$ mean noise.
When measuring the linear model is a combination of the statistical error and the systematic error so the model becomes
$$y=ax+b+\epsilon+\tau$$
$\epsilon+\tau$ is the sum of two normal variables which is itself a normal variable. These can therefore be modeled as a single random error $\zeta = \epsilon + \tau$
The mean of $\zeta$ is the sum of the means of $\epsilon$ and $\tau$, and the variance is equal to the sum of the variances. Therefore $\zeta \sim N(\mu_{sys},\sigma^2+\sigma_{sys}^2)$
So now the model is 
$$y=ax+b+\zeta$$
This is the same form as the original linear model but it has some added variance.
Both of these models are useful for answering different questions. If you take samples then you have both systematic and statistical error so you should use the latter model when performing regression.
If you have done regression and you want to predict future values then these predictions don't involve any measurement error. For this case you'll want to use the first model to exclude systematic error.
