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Among all bounded random variable $X$ in $\mathbb{R}$ such that $|X -\mu_X| \leq B$ for some $B>0$, one with the maximum variance has mass $1/2$ at $B$ and $-B$ each. Is there also a construction for the bounded RV with maximal skewness $(E[(X - \mu_X)^3])$?

I was thinking to maybe approach this as a calculus of variations problem for the pdf $f(x)$ but I'm not sure if I could do that rigorously or if it would be fruitful here. Some intuition would be great.

gwtw14
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  • Do you want the cube or the absolute value of the cube? – Ian Mar 16 '22 at 15:24
  • as you have seen, in case of the second moment one should have started not with a pdf, but just a two-point distribution. Can you find a solution for the third moment optimization for such distributions? – SBF Mar 16 '22 at 15:31
  • @Ian I believe I want the cube for the 3rd moment. – gwtw14 Mar 16 '22 at 16:36
  • @Ilya If I maximize $E[(X-\mu)^3]$ over $p$ where $X = 1$ w.p. $p$ and $X=0$ w.p. $1-p$ then the max occurs at $p = (1-\sqrt{1/3})/2$. However, how can I be sure that it is general enough to consider two-point distributions? – gwtw14 Mar 16 '22 at 16:46
  • Why the second value is $0$? You may need to optimize over it as well. As long as it’s above $-1$. Then you can use scaling argument to generalize it to all other two point distributiona – SBF Mar 16 '22 at 17:11
  • If you're considering two-point distributions, can't you WLOG choose one of them to be 0 by shifting the entire distribution, if we're only worried about centered moments? And what if I am interested in general distributions beyond 2 points? – gwtw14 Mar 16 '22 at 23:37

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