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by math_dandy
386 days ago
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This is a natural point of confusion. The true (IMO) primitive concept here is the probability measure. Probability measures on the real line are in canonical bijection with CDFs, the latter being axiomatizable as càdlàg functions (see https://en.wikipedia.org/wiki/Càdlàg) asymptotic to 0 (resp. 1) at minus infinity (resp. infinity). On the other hand, not every probability measure has a density function. (If you want the formalism of densities to capture all probability measures, you need to admit more exotic generalized functions à la Dirac.) |
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