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In a certain textbook, I see the Cumulative Distribution Function (CDF) of a continuous random variable X defined as $$\int_{-\infty}^{x'} dp(x)$$ where p(x) is the Probability Density Function of X.

Usually, I see the CDF defined thus: $$\int_{-\infty}^{x'} p(x)dx$$ I have only ever seen and solved integrals with respect to a variable, not with respect to a function of a variable, so I don't know if those two are equivalent. Are they? Or is the first expression wrong?

random_stuff
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I think you meant for the first one: $$\int_{-\infty}^{x^\prime} dP(x)$$ and the integral gives the $P$-measure of the set $(-\infty,x^\prime]$. If the probability measure is absolutely continuous wrt. the Lebesgue measure (so it has a pdf) then both definitions are equivalent but the first is more general as $P$ could have a discrete part or also a singular part.

Sergio Parreiras
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  • The textbook explicitly calls "p(x)" in the first expression "the PDF of X", but maybe that's what they meant. – random_stuff Nov 02 '14 at 19:37
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    You should change $(-\infty,x')$ into $(-\infty,x']$ here. A more rigorous notation would be $\int_{(-\infty,x]}dP\left(x\right)$. – drhab Nov 02 '14 at 19:51