Integration to d density function

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The discussion revolves around a challenging integration problem involving the expression Int(from 0 to x) (1-F(x-t)) dF(x). The user expresses confusion about calculating the integral, particularly when needing to derive results within F(x) without knowing the density function. Clarifications are made regarding the integration with respect to different variables and the implications of convolution in the context of distribution functions. Ultimately, the user finds the solution in their stochastic scriptum, revealing that for independent random variables x and y with distribution functions F and G, the distribution of their sum can be expressed as H(a)=P(x+y<=a)=int F(a-v)dG(v). This highlights the connection between integration and probability distributions in the context of random variables.
benoardo
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I have a problem with an integration, namely:

Int(from 0 to x) (1-F(x-t)) dF(x)

and do not know how to calculate...:-(
 
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If it help,s dF(x) = (dF/dx)dx
 
It helps, but that mean to multiply the integrand by the density f=dF/dx and then integrate over dx, so like how to derive the expectation...

but another problem is, that I do not know the density and need to have a result within F(x)
 
Well, you can do it, though you do'nt think you can. so let#'s play a littel game.

What's the integral of 1 with respect to x? 1 wrt y? 1 wrt F(x)?

Now what's the integral of x wrt x? y wrt y, F(s) wrt F(s)?

Note, I don't think you want an x in the integrand and in the limit.
 
That's almost all clear but in my book i read that F(s) wrt F(s) is something like F(s)*F(s) which stands here for the convolution...
And i have a x in the itegrand as well as in the limit.
As I said before: int(0 to x) 1-F(x-t) wrt F(t), sorry so you are write, it is F(t) not F(x)!
 
Thank you very much for your support, but yesterday i found the answer in my stochastic I scriptum...if someone is interested in:

Let x, y be independent rvs with dfs F and G then holds for the distribuntion of the sum x+y:

H(a)=P(x+y<=a)=int F(a-v)dG(v).
 
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