I have three random variables $X$, $Y$, $Z$. $X$ is independent from the two others. On the other hand, $Y$ and $Z$ may be dependent, and of different distributions, for example consecutive order statistics for a given sampling. How can I estimate $\mathbb P (X \in [Y, Z))$ ?
I know that this probability is $\mathbb P (\{X - Y \geq 0\} \cap \{Z-X > 0\})$, but these are not independent. Where do I go from here?
EDIT
Following @Dilip Sarwate's and @whuber's comments below. For a given $x$,
\begin{align} \mathbb P(X\in [Y, Z) | X=x) &= \mathbb E[1\{X\in [Y, Z) \} | X=x] \\ &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}1\{x\in [y, z) \}f_{Y, Z}(y, z)dydz \\ &= \int_{-\infty}^{x}\int_{x}^{\infty}f_{Y, Z}(y, z)dydz \end{align}
Then returning to the original problem:
\begin{align} \mathbb P (X \in [Y, Z)) = \int_{-\infty}^\infty\int_{-\infty}^{x}\int_{x}^{\infty}f_{Y, Z}(y, z)f_X(x)dydzdx \end{align}
Similarly, we would then have
\begin{align} \mathbb E (X 1\{X \in [Y, Z)\}) = \int_{-\infty}^\infty\int_{-\infty}^{x}\int_{x}^{\infty}xf_{Y, Z}(y, z)f_X(x)dydzdx \end{align} ?
Are the two previous equations correct?