Question

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?).

Answer 1

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) $\partial_\alpha p(q,\alpha) = - \int_{-\infty}^\infty dy \ y f(q-\alpha y,y)$.