Probability and expectation of three ordered random variables I am really stuck on the following question during my exam preparation.
Let $X_1$, $X_2$ and $X_3$ be independent exponential random variables with respective rates $\mu_1$, $\mu_2$ and $\mu_3$. Find:
a) $P(X_1 < X_2 < X_3)$
b) $E(X_2 | X_1 < X_2 < X_3) $
Question a
For question a) I've tried two approaches but they do not give me the same answer. Why the discrepancy and is one of the two even correct?
First approach
$P(X_1 < X_2 < X_3) =$
[using the law of total probability]
$ = P(X_1 < X_2 < X_3 | min X_i = X_1) P(min X_i =  X_1) +  P(X_1 < X_2 < X_3 | min X_i = X_2) P(min X_i = X_2) +  P(X_1 < X_2 < X_3 | min X_i = X_3) P(min X_i = X_3)$
[The second and third term lead to contradictions and therefor are 0]
$=  P(X_1 < X_2 < X_3 | min X_i = X_1) P(Xi = min X1)$
$= P(X_1 < X_2 < X_3, min X_i) = P(X_1 < X_2 < X_3, X_1 < X_2, X_1 < X_3)$
$= P(X_2 < X_3, X_1 < X_2, X_1 < X_3)$
[first entry contains redundant information]
$= P(X_2 < X_3)P(X_1 < X_2)P(X_1 < X_3)$
[They are independent]
$= u_2/(u_2+u_3) \cdot u_1/(u_1+u_2) \cdot u_1/(u_1+u_3)$
Second approach
The joint distribution is given by
$f(x_1,x_2,x_3) = f_1(x_1)f_2(x_2)f_3(x_3)$ due to independence
$P(X_1 < X_2 < X_3) =  \int_{x_2}^{\infty}\int_0^{\infty}\int_0^{x_2}f_1(x_1)f_2(x_2)f_3(x_3)dx_1 dx_2 dx_3 $
$= \int_0^{\infty}\int_0^{x_2}f_1(x_1)dx_1 \int_{x_2}^{\infty}f_3(x_3)dx_3 f_2(x_2)dx_2$
$= u_2/(u_2+u_3) - u_2/(u_1 + u_2 + u_3) $
Question b
For question b) I want to do the following:
$\int_{x_2}^{\infty}\int_0^{\infty}\int_0^{x_2}x_2*f_1(x_1)f_2(x_2)f_3(x_3)dx_1 dx_2 dx_3 $
am I on the right track?

HINT: 
a)
$P(X_1 < X_2 < X_3) = \int_{0}^{\infty} P(X_1 < X_2 < X_3 \ | X_2 = t)\ dP(X_2=t) = \int_{0}^{\infty} P(X_1 < t < X_3 ) \ f_{X_2}(t)\ dt$
where $P(X_1 < t <  X_3) = P(X_1<t) \cdot (1 - P(X_3 \leq t)) = (1-e^{-\mu_{1} \ t}) \cdot e^{-\mu_{3} \ t}$
b) 
$ E(X_2 | X_1 < X_2 < X_3) = E(X_2 \cdot \mathbb{1}_{\{X_1 < X_2 < X_3\}} \ ) \ / 
\ P(X_1 < X_2 < X_3)$
where $E(X_2 \cdot \mathbb{1}_{\{X_1 < X_2 < X_3\}} \ ) = \int_{x_2}^{\infty} \int_{0}^{\infty} \int_{0}^{x_2} \ x_2 \ f_{X_1}(x_1) f_{X_2}(x_2) f_{X_3}(x_3) \ dx_1 dx_2 dx_3 $
