I have essentially two questions.
How should we treat a probabilistic differential given an intergral? For example, \begin{align} \text{Compute } \int_0^a u &dF(u|x), \text{where} \\ U &\sim N(\mu_U, \sigma_U) \\ u &= e^U \\ x &= U + \epsilon, \text{where } \epsilon \sim N(0, \sigma_x) \end{align} That's a lot of stuff, but what it looks like it means is that $U$ is a random variable that in turn determines $u$ and $x$. $u$ is lognormally distributed by construction, and $x$ is constructed by adding $U$ and $\epsilon$. How do I compute the integral in this case?
In the above case, $dF(u|x)$ is the conditional probability density function (not the CDF). Therefore, \begin{align} f(u|x) = \frac{f(u,x)}{f(x)} \end{align} So if I knew the joint probability density of $u$ and $x$, and the marginal probability density of x, then I would be able to compute this. Is there a way to recover the joint probability density or the conditional probability density? I have each $f(u)$ and $f(x)$, the marginal density of each, and how they are constructed, but have no idea if I can compute the joint density (or conditional) from this.
Any advice would be deeply appreciated!