In four sections of a course, running (independently) in parallel, there are four students giving presentations that are each Exponential in length, with expected value of$10 $minutes each. How much time do we expect to be needed until all four of the presentations are completed?
I'm a little thrown off by this question since it's in the chapter of order statistics in my book. But I believe that this is just gamma distribution. If each student has expected value of $10$ minutes each. Shouldn't the time needed till all four of the presentations are completed be $40$ minutes? $(10 \cdot 4 = 40)$
Or is it the following. Calculate the density of the fourth order statistics $$f(x_4) =\frac{2}{5}e^{\frac{-x}{10}}\left(1-e^{\frac{-x}{10}}\right)^3.$$ Then $$E(X_4) = \int_0^\infty\frac{2x}{5}e^\frac{-x}{10}\left(1-e^\frac{-x}{10}\right)^3 \,dx= 125/6.$$
So is the answer $40$ minutes or $125/6$ minutes?
Any help is greatly appreciated.