Home page for accesible maths C Proofs of additional results 3.2 The Law of the Unconscious Statistician 3.4 The formula for a bivariate change of variable

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3.3 Odd central moments of symmetric distributions

Proposition 3.3.1.

For all odd $r\geq 3$ , any random variable which has a finite $r^{th}$ moment and for which the density is symmetric about the mean, $\mu$ , has an $r^{th}$ central moment of zero.

Proof.

We will use the substitutions $t=u-\mu$ and $s=\mu-u$ . Symmetry means exactly that the density satisfies $f(\mu+t)=f(\mu-t)$ for all $t$ .
Firstly

	$\displaystyle I_{1}$	$\displaystyle=\int_{\mu}^{\infty}(u-\mu)^{r}f(u)\,\mathrm{d}u$
		$\displaystyle=\int_{0}^{\infty}t^{r}f(\mu+t)\,\mathrm{d}t$
		$\displaystyle=\int_{0}^{\infty}t^{r}f(\mu-t)\,\mathrm{d}t$

by symmetry. Secondly

	$\displaystyle I_{2}$	$\displaystyle=\int_{-\infty}^{\mu}(u-\mu)^{r}f(u)\,\mathrm{d}u$
		$\displaystyle=\int_{0}^{\infty}(-s)^{r}f(\mu-s)\,\mathrm{d}s$
		$\displaystyle=-\int_{0}^{\infty}s^{r}f(\mu-s)\,\mathrm{d}s.$

Hence $\mu_{r}=(I_{1}+I_{2})/\sigma^{r}=0$ . ∎