# Are order statistics implicit in the calculation of the expected value of a bottleneck random variable?

Two points are realized on a $l\times w$ rectangle. A point has higher chance of being realized in the top-left quadrant $Q_{t,l}$. Of interest is the maximum x-coordinate among the two points. Given that one point was realized in $Q_{t,l}$ and the other not, does it suffice to find the expected value of the point that is not in the quadrant, i.e.

$$\frac{1}{3} \times \frac{l}{4} + \frac{2}{3} \times \frac{3l}{4}$$

(*) or do I explicitly need to take care of the case where the point in $Q_{t,l}$ happens to be realized at a higher x-coordinate than the point that is not in $Q_{t,l}$?

How do you solve for order statistics when there is no closed-form survival function or c.d.f. (in the variation where it is known that both points are NOT realized in $Q_{t,l}$). Opposed to the following case where there is no discontinuity: Expected value of maximum of two random variables from uniform distribution.

(*) Three quadrants remain that are not in $Q_{t,l}$. The probability that it is in the buttom-left is 1/3, 2/3 that it is one the right. If it is on the right, its expected value is halfway between l/2 and l. If it is on the bottom left, it is halfway in between 0 and l/2. Ignore the point in $Q_{t,l}$ because the expected value of the point not in $Q_{t,l}$ is father away than the expected value of the point in $Q_{t,l}$.

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It might help answer the question usefully if you describe how you arrived at the idea that it might suffice to find the expected value of the point that's not in the quadrant. –  joriki Oct 31 '12 at 4:06
@joriki: Just added it –  Wuschelbeutel Kartoffelhuhn Oct 31 '12 at 4:13
Sorry, I don't understand how what you added might suggest that you could just ignore the other point. –  joriki Oct 31 '12 at 4:15