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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 circular lake.2012-05-03

1 Answers 1

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If $(d_i)$ is nondecreasing, try $ V=\sum\limits_{i=1}^{n-1}\frac{a_i+a_{i+1}}2(d_{i+1}-d_i). $ This is as nonsensical as several other equivalent formulas, but not more.