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It is known that the probability density function $f(x)$ and the cumulative distribution function $F(x)$ are related as $f(x) = \frac{\partial F(x)}{\partial x}$. However I am confused why at some places the density function is written as just $dF(x)$.

This came up in the definition of Stieltjes Transform: $m(z) = \int \frac{1}{x - z} dF(x)$. And it is mentioned that

The density function $f(x) := dF(x)$ in the distributional sense

Is this just the issue with notation or is there specific reason to write the density function as $dF(x)$?

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There may not be an everywhere defined density function. – André Nicolas Jan 20 '12 at 20:48… – Qiaochu Yuan Jan 20 '12 at 20:49
In this instance, it is sort of short for $\frac{dF(x)}{dx} dx$ - that is, the integral is really: $\int \frac{f(x)}{x-z} dx$, but they are writing $f(x)dx = dF(x)$ as a shorthand notation and, perhaps, for clarity. – Thomas Andrews Jan 20 '12 at 21:03
@ThomasAndrews Ok. Does the term 'in distributional sense' have any specific meaning? I have also seen 'distributional derivative' term being used. Do these terms mean something? – sauravrt Jan 20 '12 at 21:46
@AndréNicolas Like atomic density functions? – sauravrt Jan 20 '12 at 21:47
up vote 0 down vote accepted

I think it's a notation thing. Maybe like the notation for a differential operator.

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This should have been a comment. – Sasha Jan 20 '12 at 21:52

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