Suppose we have two random variables, $X$ and $Z$, that have independent density functions (technically they are distributed uniformly on $[0,1]$, but for this question I don't think it matters)
My question is how to write $P(X\geq F(Z) \wedge Z\leq z^*)$ using conditional distribution.
That is, how can I write a joint density as a a product of marginals (one possibly conditional), when both parts of the joint density depend on the of the R.V.s?
(Assume $F$ is monotonic for simplicity)
Here is an illustration/ elaboration: Suppose that $X,Z$ are independent and I wish to write $P(X\geq 2-Z^2 \wedge Z\leq z^*$ as $P_{X\vert Z\leq z^*} (X\geq 2-Z^2) P(Z\leq z^*)$. Is the $P_{X\vert Z\leq z^*}$ correct, or can I write it as $P_X (X\geq 2-Z^2)$ because of independence?
Also, If i wanted to write $P_{X\vert Z\leq z^*} (X\geq 2-Z^2)$ as an integral, would it be $$1-\int_{\infty}^{z^*} P_{X\vert Z\leq z^*}(x) dx$$ This seems wrong since $z^*$ is part of the event being conditioned on, and is an upper bound (i.e. it feels like I am double counting)
Note: I think my question relates to This question, but here I am actually asking how I would write the joint probability as a product of conditional and marginal (and also how to write as an integral)