Suppose I have a r.v. $Z = X + \alpha Y$ and that $F\_Z$ is the probability distribution function of $Z$. If we think of the probability $p = F\_Z(q) = \mathbb{P}(X+\alpha Y < q)$ as a function $p = p(q, \alpha)$, how can we write the derivative $\partial\_{\alpha} p(q, \alpha)$ supposing as much regularilty on the distribution of $X$ and $Y$ as we want?


I'm loosely thinking of a situation where you know the marginals distributions $F\_X$ and $F\_Y$ and maybe locally (around $(q, \alpha)$) know some information about the dependence of $X$ and $Y$ (maybe correlation is enough in some cases to build an approximation?). 



Derivative of the CDF of a family of random variables
If $X$ and $Y$ have joint density $f(x,y)$, we have
$p(q,\alpha) = \int\_{-\infty}^\infty dy\ \int\_{-\infty}^{q - \alpha y} dx\ f(x,y)$
and thus (assuming sufficient regularity) $\partial\_\alpha p(q,\alpha) = - \int\_{-\infty}^\infty dy \ y f(q-\alpha y,y)$. 


