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When the writer wishes to show a sum/integration that applies similarly to both pmfs and pdfs, they write it like this:

$\quad \int (whatever)\ dF_X(x) $

Such that this becomes either $\sum_x (whatever) \ f_X(x)$ or $\int (whatever)\ f_X(x)\ dx$ depending on whether $X$ is discrete or continuous.

What is the intuition behind $dF_X(x)$ here? (Integrating by the cdf? But how/why?) I know what it turns into, but I don't know why.

kjetil b halvorsen
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Stats newb
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    All (Lebesgue–Stieltjes ) integration is with respect to a measure. A discrete distribution uses the counting measure, which when the integral is evaluated, is a discrete summation. – Mark L. Stone Oct 31 '15 at 22:27
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    Thanks for the answer. I don't think I follow, though - could you assume that I am a little dumber? Or maybe point me to a resource that gives another example of this notation? – Stats newb Oct 31 '15 at 23:16
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    Try reading http://artsci.wustl.edu/~econ508/2_MeasureTheoryBasics.pdf . I don't promise it will be easy, but this seems easier to understand than other things I found by googling. – Mark L. Stone Oct 31 '15 at 23:55
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    The same notation is used for both [Stieltjes integration](http://mathworld.wolfram.com/StieltjesIntegral.html) and Lebesgue-Stieltjes integration. Kendall and Stuart (*Advanced Theory of Statistics*, particularly vol I, very readable despite the title) briefly discuss the former (and use it throughout) it's a precursor to the [latter](https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration), which is more general. It's not just doing discrete and continuous at the same time, though. e.g. consider trying to write down (say) the integral for a variance for a mixed-type distribution – Glen_b Nov 01 '15 at 01:13
  • See this: https://stats.stackexchange.com/questions/268024/choosing-the-number-of-bootstrap-resamples/268162#268162 for an example showing the utility of using the Stieltjes integral notation. Your question is answered multiple times at math SE, as: https://math.stackexchange.com/questions/1581743/what-is-the-difference-between-the-riemann-integral-and-the-stieltjes-integral-d/1581755#1581755 – kjetil b halvorsen Jul 31 '18 at 12:23

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