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We were asked to use the Maclaurin series expansion of the moment generating function of the standard normal distribution. Please explain why the rth moment about the mean is 0 when r is odd. Thank you

ANONYMOUS
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  • I already did the expansion. It's just not clear to me why it's 0 when it's odd. – ANONYMOUS May 17 '18 at 17:00
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    if $\mathbb{E} e^{i\lambda X}$ is your moment-generating function, then expansion gives you a power-series in $\lambda$. On the other hand, you can compute the moment-generating function explicitly using $X\sim N(0,1)$. That will give you another function in $\lambda$ which you can expand as a power series. You get 2 different power series in $\lambda$ which must coincide. Hence all their coefficients must be the same. Doing the comparison of coefficients of $\lambda^n$ on both sides, gives you the result. – Hayk May 17 '18 at 17:27
  • Are you asking why the expansion of $\exp(t^2/2)$ only has non-zero coefficients for even powers? Or why a distribution symmetric about zero has odd moments which are zero? – Henry May 17 '18 at 23:11

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