Let $X$ be uniformly distributed with mean zero, $X \sim \mathcal{U}(-a,a)$, and a cumulative distribution function (CDF) denoted $F$.
I am interested in finding the probability $P(X^2 - X<c)$, where $c$ is a constant.
I know that I would need the joint CDF in order to compute it. I can arrive at the distributions for $X^2$, which is not uniformly distributed. The CDF for $X^2$,denoted $G(x)$, that I obtain is $G(x)=F(\sqrt{x}) - F(-\sqrt{x})$ and of course from there I can arrive at the pdf.
I am aware that in general it is typically not possible to infer the joint distribution from the marginals in the case of dependent random variables, so at this point I am kind of stuck. As far as I know, in general there could be infinitely many joint distributions that fit two marginals.
Nevertheless, is it possible in this case of uniform distribution to solve for the joint CDF?
Hint: Consider the event $X^2 - X<c$. See that this corresponds to $ x_1 \le X \le x_2$, where $x_1,x_2$ are the roots of $X^2 - X =c$. You should be able to compute this probability in terms of $F_X$ (or simply use the information that $X$ is uniform...)