Four students are giving presentations

Question

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.

They are happening simultaneously, so you are seeking the expected maximum time, not the total time. — Thomas Andrews, May 15 '21 at 23:58
I don’t know, I can’t follow your argument because I don’t know order statistics, and the formatting is nasty. — Thomas Andrews, May 16 '21 at 00:06
Oh. What function is better to use? I'm still relatively new to mathjax? — , May 16 '21 at 00:11
Just plain \frac. I’ve edited to make it clearer. I often prefer $$e^{-x/10}$$ rather than any fraction formatting. It is often easier to read, in my opinion. — Thomas Andrews, May 16 '21 at 00:13
The formula in the Wikipedia article on order statistics gives $125/6$ — saulspatz, May 16 '21 at 00:45

score 0 · Answer 1 · answered May 16 '21 at 00:39

0

Take a look at this question. You are seeking $\mathbb{E}[\max\{X_1,X_2,X_3,X_4\}]$.

answered May 16 '21 at 00:39

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Four students are giving presentations

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