# How to calculate the cumulant of a generic function of random variables?

Consider the example: Let $X$, $Y$ be two joint random variables and $\alpha$, $\beta$ two parameters, then show that $$\langle X {\rm e}^{\alpha X + \beta Y}\rangle = \langle \langle X {\rm e}^{\alpha X + \beta Y} \rangle\rangle\langle {\rm e}^{\alpha X + \beta Y} \rangle$$ where $\langle .\rangle$ indicates moment (average) and $\langle\langle . \rangle\rangle$ cumulant.