I have a simple model that says $y = c_1x^2 + c_2x$ and I have a standard deviation for the coefficients $c_1$ and $c_2$ which are normally-distributed and independent.
How do I determine the standard deviation of $y$ at different values of $x$? The analytical solution isn't clear to me. What I have done is run a monte carlo simulation where I do multiple draws for $c_1$,$c_2$ and I just evaluate $y$ many times and obtain the resulting standard deviation of this data.
How do I go about finding the answer without doing the simulation? I have read up on error propagation, but I'm not sure that it applies. Many thanks for the help.
this is a sum of independent random variables so you can apply the formula $\text{Var}(ax+by) = a^2\text{Var}(x)+ b^2\text{Var}(y)$.
In your case
$$\text{Var}(y|x) = x^4\text{Var}(c_1)+ x^2\text{Var}(c_2)$$