There's two lamps in a room.
One from type $A$, and one from type $B$.
Let $T_A$ be the lifetime of lamp $A$ (in months), and $T_B$ the life time of lamp $B$ (in months).
PDF's of $T_A,T_B:$
$f_{T_B}=3e^{-3t}, t\ge 0$. and otherwise $0$.
$f_{T_A}=e^{-t}, t\ge 0$ . otherwise $0$.
basically $T_A\sim exp(1), T_B\sim exp(3)$
The question was: We lit the two lamps together at the start of the month, what is the expected value of the time from the start of the month until the last lamp burns?
The solution did this:
$X=max\{T_A,T_B\}$.
And so $E(X)=\int_0^{\infty}P(X>x)dx=\int_0^{\infty}(1-P(max\{T_A,T_B\}\le x)=\int_0^{\infty}(1-P(T_A\le x,T_B\le x)dx$.
And then they did this:
$\int_0^{\infty}1-P(T_A\le x)P(T_B\le x)dx$.
I know that this comes from $T_A,T_B$ being independent, but the question doesn't say so, is there anything I'm missing in this question that tells me they're independent? How would I check that?
Any help is really appreciated, thanks in advance.