5.3.2 problem 39

Let $O_1,O_2,O_3$ be the offers received, all distributed as $\mathrm{Expo}(1∕12000)$. We want to find $E\left(\max \right(O_1,O_2,O_3\left)\right)$. Imagine ordering the offers as lowest, middle, and highest price. Let $D_1$ be the lowest price, let $D_2$ be how much more the middle price is than the lowest price, and $D_3$ be how much more the highest price is than the middle price. Then $D_1$ is the minimum of 3 $\mathrm{Expo}(1∕12000)$ variables (since it is just the minimum of the 3 offers), $D_2$ is the minimum of 2 $\mathrm{Expo}(1∕12000)$ variables (by the memoryless property, given that all other offers are at least the lowest offer, the likelihood that any offer is at least $k$ more than the lowest is just the probability the offer was at least $k$ in the first place), and similarly, $D_3$ is the minimum of 1 $\mathrm{Expo}(1∕12000)$ variable. In addition, we realize that $D_1+D_2+D_3$ must equal the highest offer.

Then $E(D_1+D_2+D_3)=\frac{1}{3\lambda }+\frac{1}{2\lambda }+\frac{1}{\lambda }=4000+6000+12000=22000$.