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A distribution is a gadget that tells you how much there is over any region of space.

A density is a gadget you have to integrate in order to find out how much there is over any region of space.

Incidentally, in the probability case, we will often define the distribution in terms of a "cumulative distribution function": we use p(x) to tell how much there is over the interval (-infinity, x).

(Or maybe it's (-infinity, x]. I can't remember)

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