There does not seem to exist a literature reference for the energy functionals of the log-normal distribution, we should try to derive it, or find a reference.
Collecting discussion below, from #214.
Current state:
- explicit formula for the cross-term $\mathbb{E}[|X-c|]$ (almost) derived
- no progress yet on the self-term $\mathbb{E}[|X-X'|]$
@bhavikar04 used Wolfram Alpha to derive the following indefinite integral related to the cross-term $\mathbb{E}[X-c]$:
My reply:
this looks correct. Now you need to add the limits. That should be an easy substitution, no? I recommend, do that manually. Use that
$\lim_{x\rightarrow -\infty} \mbox{erf}(x) = -1$, and $\lim_{x\rightarrow \infty} \mbox{erf}(x) = 1$. You need to be careful with the sign, but that should be it?
The number 0.707 etc should be $\frac{1}{2} \sqrt{2}$, but it doesn't matter for the limits.
There does not seem to exist a literature reference for the energy functionals of the log-normal distribution, we should try to derive it, or find a reference.
Collecting discussion below, from #214.
Current state:
@bhavikar04 used Wolfram Alpha to derive the following indefinite integral related to the cross-term$\mathbb{E}[X-c]$ :
My reply:
this looks correct. Now you need to add the limits. That should be an easy substitution, no? I recommend, do that manually. Use that
The number 0.707 etc should be$\frac{1}{2} \sqrt{2}$ , but it doesn't matter for the limits.