2
$\begingroup$

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 $dF(x)$?

  • 0
    There may not be an everywhere defined density function.2012-01-20
  • 0
    http://en.wikipedia.org/wiki/Distribution_(mathematics)#Differentiation2012-01-20
  • 0
    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.2012-01-20
  • 0
    @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?2012-01-20
  • 0
    @AndréNicolas Like atomic density functions?2012-01-20
  • 0
    Yes, presumably the integral is the Stieltjes integral.2012-01-20
  • 0
    It's a Stieltjes integral wrt $F(x)$ and $dF(x)=f(x)$ in the sense that if a density $f$ with respect to some dominating measure $\mu$ exists then $\int g(x) \ dF(x) = \int g(x) f(x) \ \mu(dx)$ meaning that if one side of this equation is defined then so is the other and they are equal. I probably would have wrote $dF(x) = f(x) \ \mu(dx)$ instead.2012-01-22

1 Answers 1

0

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

  • 0
    This should have been a comment.2012-01-20