I have the following integral: $$ \int_{0}^{n} (a x+b) g(x) dx $$ $g(x)$ is a probability density function. and $x= \epsilon + c$ where $c$ is deterministic and $\epsilon \sim f$. I want to rewrite the integral in terms of $\epsilon$ and $f$.
is this correct, what is the general rule, what is called? $$ \int_{-c}^{n-c} (a (\epsilon + c)+b) f(\epsilon) d\epsilon $$
As of your edit that removed $\epsilon$ from the first integral's upper limit, that $\epsilon$ is random became irrelevant. If you transform the integral with the substitution $x=\epsilon+c$, the result is $\int_{-c}^{n-c}(a(\epsilon+c)+b)g(\epsilon+c)d\epsilon$. If $g$ is the PDF of $X$, $\epsilon$ has PDF $f(\epsilon)=g(\epsilon+c)$, so the integrand's $g(\epsilon+c)$ factor can be replaced with $f(\epsilon)$. There's no rule name to learn here. But we can make this now intuitive by writing the result as the trivial $\Bbb E(aX+b)=\Bbb E(a(\epsilon+c)+b)$.