Emails come from two professors at the rate of 4/hour and 6/hour respectively. What is the expected time I will have to wait until both the emails arrive?
In this question, am I supposed to find maximum time for (X,Y) where X represent time of arrival of email from professor A(say) and Y represents time of arrival of email from Professor B?
Hint: "$\max(X,Y)\le t$" is the same as "$X\le t$ and $Y\le t$". Therefore $$ P(\max(X,Y)\le t)=P(X\le t,Y\le t)=P(X\le t)P(Y\le t),\tag1 $$ the last equality by (assumed) independence. Now you can deduce the density of $\max(X,Y)$ from (1), and then compute the expectation.
Another approach: use the identity $$ E(T) = \int_0^\infty P(T>t)\,dt, $$ which is valid for any nonnegative random variable $T$.