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Imagine a probability distribution. Now imagine its density or mass function. Imagine that the graph of the function is actually printed out on a piece of paper (possible an 'infinitely big' one) - or actually, on a thin wooden board. Now imagine cutting out the bit of the wooden board that is inside the function, leaving a thin sliver along the x-axis wherever needed in order to connect the whole function together into one cutout.

Lastly, imagine trying to balance this wooden cutout of the p{m,d}f on a point. Is the point on which it would balance always the expectation of the distribution?

I think of course you'd always say that the 'along the x-axis' bits have weight 0, even though in real life they would of course have a weight.

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I just noticed that when you have a pmf this isn't a very helpful intuition, because each mass spike/bar has width '0' and I don't think humans have very good intuitions about balancing objects made up of a bunch of parallel spikes. But it seems helpful for pdfs. – Marius Kempe Apr 8 '12 at 22:20
A spike in the pmf is a point mass. That's not too hard to imagine; you just think of putting a small heavy lead weight on top of your infinitely light wooden board. – Rahul May 8 '12 at 19:33
up vote 2 down vote accepted

Nice observation! Yes, expectation in Probability Theory and (first) moment in Physics are mathematically the same thing. The analogies extend further. Second moments are important both in Probability Theory (variance) and Physics (moment of inertia).

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