# Differentiation of expectation of a nonlinear function

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!

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I think from what you specify the problem can't be reduced further than this:

$$\begin{eqnarray} \frac{\partial}{\partial\alpha_i}E\left(G(Y)\right) &=& E\left(\frac{\partial}{\partial\alpha_i}G(Y)\right) \\ &=& E\left(\frac{\partial}{\partial\alpha_i}\int_0^Yc(x)\mathrm{d}x\right) \\ &=& E\left(c(Y)\frac{\partial Y}{\partial\alpha_i}\right) \\ &=& E\left(c\left(\sum_{j=1}^N \alpha_j X^{\kappa_j}\right)X^{\kappa_i}\right)\;. \end{eqnarray}$$

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