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I have a variable that is distributed according to a lomax distribution (https://en.wikipedia.org/wiki/Lomax_distribution) with CDF and PDF given by

$F(x) = 1-\left(1+x\right)^{-\alpha}$

$f(x) = \alpha(1+x)^{-\alpha-1}$

I know that moments are given by

$\int_{0}^\infty x^n f(x) dx = \frac{\Gamma(\alpha-n)\Gamma(1+n)}{\Gamma(\alpha)}$

I want to compute partial moments of the form

$\int_{y}^\infty x^n f(x) dx$

but I am having trouble figuring out how to derive an expression for the partial moments. Can anyone give me a hint? Thanks!

j steinberg
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  • Your issue is equivalent to find an antiderivative of $x^nf(x)$. It is not sure at all that a close expression exists for it. – Jean Marie Mar 28 '19 at 19:43
  • I am aware of that, and I am guessing there is no closed-form antiderivative. But since someone has figured out how to express the non-partial moments in terms of gamma functions (which are easy to compute), I was hoping to get something similar for the partial moments. Perhaps it's impossible. – j steinberg Mar 28 '19 at 19:48
  • Let me add to my previous comment. I do not necessarily need a closed-form solution for the partial moments; I simply need a way to compute them easily. – j steinberg Mar 28 '19 at 19:57

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