# Probability of a random variable being between two random variables

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?

• Try finding the conditional probability that $Y \leq X$ and $Z > X$ conditioned on $X$ having taken on the value $x$. Then, remove the conditioning on $X$ by multiplying your result by the pdf $f_X(x)$ of $X$ and integrating. – Dilip Sarwate Sep 11 '19 at 3:23
• (1) How might $X$ be related to $Y$ and $Z$? Same distribution? Independent? (2) What information or data would you consider using to estimate this probability? Or do you really mean to ask how to compute its value? – whuber Sep 11 '19 at 15:57
• @whuber, I edited the question: $X$ is independent from the others. The idea would be to have some bound on some distance in probability space between the distribution of $X$ and $Y, Z$. – Geoffrey Negiar Sep 11 '19 at 16:12
• Does $X$ have the same distribution as $Y$ and $Z$ or not? It would also be useful to indicate whether any of these distributions might not be continuous. – whuber Sep 11 '19 at 16:14
• It doesn't necessarily have the same distribution. We can suppose all of them to be continuous. – Geoffrey Negiar Sep 11 '19 at 18:44

$$\mathbb P (X \in [Y, Z)) = \int_{-\infty}^{\infty} \mathbb P (x \in [Y, Z)) f_X(x) dx = \int_{-\infty}^{\infty} (\underset{(x, \infty) \times (-\infty, x)}{\iint} f_{YZ}(y, z) dydz) f_X(x) dx$$
since $$Y, Z$$ are dependent can you substitute double integral with two integrals? also it may be easier to compute this integral