I am having some problems working with stochastic processes/random variables whose probabilities are themselves random. In particular I am not really sure which "operations" are allowed and how to think of them.
For instance, suppose $A,B$ are two events, but suppose that $P(A)=X_A$ and $P(B)=X_B$ where the $X_i$ themselves are random variables. Now, if I wanted to compute $P(A\cap B)$ I would probably want to use itterated expectation. But its not really clear to me how to do this. I mean $P(A\cap B)$ should be a random variable, but $E[P(A\cap B|\mathcal{F})]$ isn't right? So something goes wrong here and it appears this doesn't work, at least not in a totally straight-forward manner. Is there any way around this?
A more sophisticated example would be a Poisson Process where the intensity itself is random. I.e. suppose $N(\cdot)$ is a Poisson Process with intensity $\lambda$. Now assume that Poisson Process is "thinned doubly randomly". That is, we keep each point $i$ of the process with probability $X_i$, which itself is a random variable on $(0,1)$. Denote the thinned process by $M(\cdot)$. Now, to characterize this process, I want to compute $P(M(A)=k)$ for some borel set $A$. How is this probability interpreted? Is it a random variable, and if yes, can we write: \begin{equation} P(M(A)=k)=\frac{e^{-\lambda X}(\lambda X)^k}{k!}? \end{equation} Moreover, if this is the case, how do I compute this?
Also, I haven't found much on the topic in my standard probability references. If anyone has good literature on this, it would be hugely appreciated!