# What is the formula of the linear regression with an error propagation

I am in Physics Licenciature and a day the teacher showed me a formula for the linear regression with error propagation, and time after, I was searching this formula and I didn't find it. Then I am frustrated, so any answer will be well received.

• Does this link help at all en.m.wikipedia.org/wiki/Propagation_of_uncertainty ? It was, by the way, the first hit on a google for “propagation of error”. – Nap D. Lover Jan 6 at 23:50
• I know the propagation of uncertainty and its calculus, but I don't know the uncertaintly of the linear regression of various data – El borito Jan 6 at 23:55
• Well, it is not very clear what you are asking for anyway. Are you asking: if $Y_n =\alpha +\sum_{i=1}^N \beta_i X_{n,i}+\epsilon_n$, (i.e. a linear regression model with an intercept and $N$ independent variables) and $Var(X_{n,i})=\sigma_i^2$ then what is $Var(Y_n)$? If so, then this is indeed answered already in the linked article I provided (up to some assumptions on $\epsilon_n$)... – Nap D. Lover Jan 7 at 0:08
• It's not what I was looking for, but in fact an answer like that would be very useful. – El borito Jan 7 at 0:17
• Hmm, perhaps you should ask your lecturer then? Anyway, the result I eluded to in my previous comment is just an application of the general formula for variance of a sum of (possibly correlated) RVs: $$Var(X_1+\dotsc +X_n)=\sum_{i=1}^n Var(X_i)+2\sum_{i <j} Cov(X_i X_j).$$ – Nap D. Lover Jan 7 at 0:26