The Problem with Odd Moments
Last week, realizing that it had been a while since I posted anything in the cute problems folder, I did a quick post before going to a meeting. Unfortunately, as a couple of people pointed out almost immediately, there was a problem with the question (a typo in the form of a misplaced bracket). I took the post offline until I could correct it and then promptly forgot about it. I remembered it yesterday so have now corrected it. I also added a useful integral as a hint at the end, because I’m a nice person. I suggest you start by evaluating the expectation value (i.e. the first-order moment). Answers to parts (2) and (3) through the comments box please!
Answers to (2) and (3) via the comments box please!
SOLUTION: I’ll leave you to draw your own sketch but, as Anton correctly points out, this is a distribution that is asymmetric about its mean but has all odd-order moments equal (including the skewness) equal to zero. it therefore provides a counter-example to common assertions, e.g. that asymmetric distributions must have non-zero skewness. The function shown in the problem was originally given by Stieltjes, but a general discussion can be be found in E. Churchill (1946) Information given by odd moments, Ann. Math. Statist. 17, 244-6. The paper is available online here.
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In the Dark 
July 9, 2018 at 2:01 pm
So, a distribution defined from minus infinity to plus infinity that is not symmetrical about the origin yet still has all of its odd moments zero. That is possible only because it has no power-series expansion, of course.
July 10, 2018 at 3:40 pm
I worked out the answer to this in the shower this morning.
I didn’t do the integrals in my head — my integration skills aren’t bad, but they’re not that good! Rather, I figured out what the answers must be based on the fact that you’d asked the question. The only answers to (1) and (2) that would be surprising enough to lead you to pose the question are an asymmetric distribution with all odd moments zero.
Actually, I guess the reverse — an even function with nonzero odd moments — would also be surprising enough! In fact, it would be a good deal too surprising, since such a thing really is impossible. (You could certainly construct one where the integrals for the moments failed to converge, but not one where they converge and are nonzero.)
Is it really a common assertion that an asymmetric distribution must have nonzero skewness? I’d hope that people could spot that that one is false without too much effort. (You can construct discrete counterexamples with as few as three possible outcomes.) An asymmetric distribution with *all* odd moments equal to zero is considerably more surprising.
July 10, 2018 at 3:55 pm
There are lots of examples of discrete distributions with this property.
I have heard quite a few people in talks state that asymmetric distributions have to have non-zero skewness, which is why I hunted out this example.