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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?

Four Seasons
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    Yes, that is what you are supposed to find: the expected value of the maximum of the two exponentially distributed random variables. (At least, that's the way I read the question.) – Brian Tung Apr 30 '18 at 20:44
  • Ok, in that case, if I define Z (a new random variable) to be max(X,Y), then I found that Z is also exponentially distributed with rate 10/hour. What next, then? – Four Seasons Apr 30 '18 at 20:50
  • No, the minimum should be exponentially distributed with rate $10$. The maximum is not exponentially distributed. – Brian Tung Apr 30 '18 at 21:26
  • Ok. Any hint on how I should proceed? – Four Seasons Apr 30 '18 at 21:34

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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 $T$.

grand_chat
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