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Let $X\geqslant 0$ be a random variable. Then, we have

$$\mathcal{E}(X)=\int_0^\infty P(X>t)dt$$

(provided $\mathcal{E}(X)$ exists).

Suppose we have a finite data set $\{(d_1, a_1), \ldots, (d_n, a_n)\}$ consisting of pairs or real numbers where $d_i$ stands for a level (height) of some vessel and $a_i$ is the area of the surface of the vesel at level $d_i$.

How can I apply the above mentioned formula to calculate the (expected) capacity of the vessel?

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    A single vessel? Anything known about the shape? If the vessel is a conical cup, the measurements can tell us nothing about *capacity*. The cup could be very tall, but our measurements might involve only small amounts of water.2012-05-03
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    The vessel is a lake, essentially.2012-05-03
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    A circular lake.2012-05-03

1 Answers 1