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Is there any nice geometric interpretation of the mathematical expectation of a random variable (preferably based on density or cumulative density plot)?

(For example, median has a nice geometric interpretation as the point dividing the area under the density curve into two parts of equal size. Meanwhile, mode corresponds to the highest point of the density. But what about the mathematical expectation?)

Richard Hardy
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  • I searched for the answer on Cross Validated first, could not find anything, so posted the question. Then I remembered that statistics resources exist in other places besides Cross Validated, too, so I checked Wikipedia and rather quickly found what I needed (posted as an answer). – Richard Hardy Sep 10 '18 at 12:25
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    I found a nicely illustrated answer at https://stats.stackexchange.com/questions/132914/what-exactly-are-moments-how-are-they-derived/132968#132968 by searching for ["moment interpr*".](https://stats.stackexchange.com/search?q=moment+interpr*) – whuber Sep 10 '18 at 14:34

1 Answers1

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The mathematical expectation is the x-coordinate of the centre of gravity.

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The picture above is borrowed from Wikipedia.

Richard Hardy
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    Is this *not* (?) nice enough: the expectation as a weight balancing point? – Jim Sep 10 '18 at 10:27
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    @Jim, Yes, it is, and *therefore* I posted this answer. I searched for the answer on Cross Validated first, could not find anything, so posted the question. Then I remembered that statistics resources exist in other places besides Cross Validated, too, so I checked Wikipedia and rather quickly found what I needed. – Richard Hardy Sep 10 '18 at 10:29