Joint and Marginal Probability Densities

Question

I have essentially two questions.

1. 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?

2. 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!

Answer 1

Since $x$ is the sum of two (presumed) independent normal distributions, $U$ and $\epsilon$, it is also normally distributed:

$$x\sim \mathcal N(\mu_U, \sigma^2_U+\sigma^2_x)$$

$$f_x(\xi) = \frac{\phi\left(\tfrac{\xi-\mu_U}{\sqrt{\sigma^2_U+\sigma^2_x}}\right)}{\sqrt{\sigma^2_U+\sigma^2_x}}$$

The joint density of $u,x$ is related to the joint density of $U$ and $\epsilon$ by a change of variable transformation.

$$\begin{align}f_{u,x}(\eta,\xi) = & ~ \frac 1{\lvert \eta\rvert} f_{U,\epsilon}(\ln\eta,\xi-\ln\eta) \\[1ex] = & ~ \frac{\phi\left(\frac{\ln(\eta)-\mu_U}{\sigma_U}\right)~\phi\left(\frac{\xi-\ln\eta}{\sigma_x}\right)}{\lvert \eta\rvert ~\sigma_x ~\sigma_U}\end{align}$$