Probability in train station So I have this problem that I'm thinking about and I don't really know how to solve. So we are in a train station and we ask a random person when is the next train in a certain direction going to come. He says that there are 2 trains coming. He doesn't know which one will arrive first. What he knows for sure is that train "A" will arrive in maximum 10 minutes and that train "B" will arrive in maximum 8 minutes. The probability density for each train arrival is constant during their time intervals. What is the mean time interval until any train arrives?
Thanks! 
 A: $T_1$ arrival time of train $A$,  uniform on $[0,10]$. 
$T_2$ arrival time of train $B$,  uniform on $[0,8]$. 
$T= \min (T_1,T_2)$ is the time of arrival of the first train. 
To calculate the expectation: 


*

*Recall that for a nonnegative random variable $X$, $E[ X] =\int_0^\infty P(X>t) dt$.

*By definition of $T$ and independence of $T_1$, $T_2$, 
$$(*)\quad P(T>t) = P(T_1>t \mbox{ and }T_2>t) = P(T_1>t) P(T_2>t)=\frac{10-t}{10}\times \frac{8-t}{8},$$
if $t \in [0,8)$, and $P(T>t)=0$ for $t\ge 8$. 


(note: you can differentiate this to obtain the density of $T$ and use it to calculate the expectation). 
Let's put all of this together: 
$$E [T] = \int_0^{\infty} P(T>t) dt = \int_0^{8} P(T_1>t)P(T_2>t) dt = \int_0^{8} \frac{10-t}{10}\times \frac{8-t}{8} dt.$$
A: Let $x$ be the time until train A arrives and $y$ be the time until train B arrives.  You can represent the arrival times as a random point in the rectangle $0 \le x \le 10, 0 \le y \le 8$.  The chance that the first train arrives in the time interval $t$ to $t+\Delta t$ is union of two strips:  $(t \le x \le t+\Delta t) \times (t \le y)$ and a corresponding horizontal one.  This gives you the pdf of $t$.
