Integration to d density function

[benoardo]

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...:-(

 

[matt grime]

If it help,s dF(x) = (dF/dx)dx

 

[benoardo]

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)

 

[matt grime]

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.

 

[benoardo]

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)!

 

[benoardo]

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).