# Is the expectation of a probability distribution the point on which a physical cutout of the distribution would balance?

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