I encountered the following problem and need some help.
Let $X$ be a continuous random variable. (You can assume $X$ to be very nice: it has a smooth density function with bounded support, bounded away from 0.) Define $Y=\sum_{i=1}^N \alpha_i X^{\kappa_i},$ where $\alpha_i>0, \kappa_i<0$ are parameters. What I need is to compute $\frac{\partial}{\partial \alpha_i} \mathbb{E}(G(Y)),$ where $G$ is a nonlinear function. (If it helps, $G(y)$ is given by an integral $G(y)=\int_0^y c(x)dx$ with $c(x)$ nice.)
Thank you!