Probability that a random variable lies between two different random variables. Say we have three different random variables $X_{1}$, $X_{2}$, and $X_{3}$ with pdf's $f_{X_{1}}$, $f_{X_{2}}$, and $f_{X_{3}}$. Random variable $X_{2}$ is independent of random variables $X_{1}$ and $X_{3}$. But random variables $X_{1}$ and $X_{3}$ are not independent of each other. What is the probability that $X_{1}$ < $X_{2}$ < $X_{3}$.
My solution:
Pr{$X_{1}$ < $X_{2}$ < $X_{3}$} = Pr{$X_{2}$ < $X_{3}$} - Pr{ $X_{2}$ > $X_{1}$}. From here on, I take the standard approach. Is this correct?

No! I don't know how you arrived at that equation. The correct method is this: $P\{X_1<X_2<X_3\}=E\int_{X_1}^{X_3} f_{{X_2}}(x) \, dx=\int \int\int_u^{v} f_{{X_2}}(x) \, dx \, f_{{X_1},{X_3}}(u,v) \, du \, dv$ which depends on the joint distribution of $(X_1,X_3)$. You cannot express this in terms of the marginal densities $f_{{X_1}}$ and $f_{{X_3}}$.
