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I have a very basic question I couldn't find the answer of after a quick Google search: If the expectation of a random variable is 0, is the expectation of the cubed random variable also 0? Is the expectation of the random variable to the power of any odd number 0 in this case? It makes sense to me, but maybe there is a counter example. For normal distributions with zero mean the odd moments are all 0, but I wanted to know if it's the case for other distributions as well.

Thank you very much in advance.

Jose Avilez
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Leo
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No, take $X$ having support $\{-1,2\}$ with respective probability masses $2/3,1/3$. Then $X$ has mean zero but third moment

$$(-1)^3(2/3)+2^3(1/3)=2.$$

Golden_Ratio
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  • And for continuous distributions there are also counter-examples? – Leo Feb 17 '22 at 23:21
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    @Leo of course. Try this: support $[-1,2]$ with density $f(x)=4/9-(2/9)x$ – Golden_Ratio Feb 18 '22 at 00:11
  • Take any normal distribution with non-zero mean $\mu$ and arbitrary standard deviation $\sigma$. Then $E(X) = \mu$ and $E(X^2) = \mu^2+\sigma^2$, while, $E(X^3) = \mu^3 + 3\mu\sigma^2$. – user170231 Feb 18 '22 at 00:13
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    @user170231 But OP wants zero mean and nonzero third moment... – Golden_Ratio Feb 18 '22 at 00:16
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    @Leo As for why odd moments of a mean zero normal random variable are zero, it has to do with symmetry. Random variables having densities symmetric about zero have odd moments equal to zero: https://math.stackexchange.com/questions/72451/odd-order-moments-of-a-symmetrical-distribution – Golden_Ratio Feb 18 '22 at 00:19