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I am looking for a distribution with a differentiable PDF $f:(0,\infty)\rightarrow (0,\infty)$ for which for any $\delta>1,z>0$, the two following integrals are finite elementary functions of $\delta$ and $z$: $$\int_0^z{x^{\delta} f(x)\,dx}$$ $$\int_z^{\infty}{x^{-\delta} f(x)\,dx}$$

The distribution's CDF ($\int_0^z{f(x)\,dx}$) should also be elementary and its first moment $\int_0^{\infty}{x f(x)\,dx}$ should be finite.

If these integrals started at $1$, not $0$, then the Pareto distribution would satisfy all requirements. And if I dropped the differentiability requirement, then a distribution with PDF proportional to $\exp{[(\alpha+1)|\log x|]}$ would work (with $\alpha>0$). Is there a differentiable solution?

Note: The original version of this question talked about tractability rather than being elementary. My definition of tractability was a bit vague, and was too loose for what I actually need.

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    $\begingroup$ I haven't checked all the requirements, but the exponential could be a good starting point for tractability: docs.scipy.org/doc/scipy/tutorial/stats/… Are you in fact looking for a truncated distributions? Your formula suggest (to me) that you consider the regular density, but not integrated over the entire support. $\endgroup$
    – Christoph Hanck
    Commented Feb 15 at 16:45
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    $\begingroup$ @ChristophHanck Thanks. Maple had given the integral in the exponential case in terms of the WhittakerM function, but I now see that doing a "convert(%,GAMMA_related)" produces the more reasonable looking expression you linked to. I expect a similar expression in terms of the bivariate Gamma exists for many of the other distributions I mentioned, as the exponential is a special case of them. There is no real difference between the truncated moments being tractable and what I asked for, since the truncated moments are tractable if the CDF and the integrals I gave are tractable. $\endgroup$
    – cfp
    Commented Feb 15 at 17:32
  • $\begingroup$ @ChristophHanck So, the Gamma, Inverse Gamma, Frechet and Weibull distributions all work, contrary to what I wrote in the question! In each case, the WhittakerM function that Maple produced could be converted to a bivariate Gamma function. Thanks again for the tip! $\endgroup$
    – cfp
    Commented Feb 15 at 17:50
  • $\begingroup$ I have substantially revised the question, as the original notion of tractability was too loose. For example, I said the error function was fine, but I was forgetting that while these moments are easily derived for the log-normal distribution, they were still not tractable enough for my purposes. $\endgroup$
    – cfp
    Commented Feb 16 at 10:26
  • $\begingroup$ @cfp: From the comments it looks like you have answered your own question, so maybe make it into a formal answer here? $\endgroup$
    – kjetil b halvorsen ♦
    Commented Feb 16 at 16:43

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