Is "Cumulative CDF" a studied topic?

Question

I am wondering if there is a special term used for the "second integral of pdf function" which is the same as the "integral of cdf"? or also as "cumulative cdf"? Does a branch of math study this function?

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Update: A suggested answer is "expected cdf", but I do not think what I am describing is the expected cdf (even upto constant scale). To see it, let a pdf be noted with $()$ and cdf be noted with $()$. Then the cumulative CDF which I look for is the function $$ℎ_1(_0)=\int_{-\infty}^{x_0}(),$$ while an expected cdf is $$ℎ_2(_0)=\int^{_0}_{−∞}()().$$ Unless someone claims the integrals are the same by some change of variable?

Answer 1

integral of pdf is the cdf since it represents the distribution of the "density" of probability, or at which domain the event is more likely to happen. Well for cumulative CDF, it has something to do with expectation, you can read "A first course in probability" by Dr. Ross, 9th edition. In page 181, Chapter 5, see Lemma 2.1.

"For a nonnegative random variable $Y$, $$E[Y]=\int_{0}^{\infty}P\{Y>y\}dy$$ "

Here, $P\{Y>y\}$ can be replaced by $1-F(y)$, which is a function of cdf.

Answer 2

Integrated distribution functions are, for example, studied in:

https://arxiv.org/pdf/1801.00977.pdf

Answer 3

Interesting! The equation $$E[x] = \int_0^{+\infty} 1- F(x)\quad dx$$ mentioned by @X.Kong above is very useful and not trivial at all. It defines a deep relationship between the integral of a CDF and an expected value.