TITLE: Is the kth central moment less than the kth raw moment for even k?
QUESTION [6 upvotes]: If $X$ is a real-valued random variable, then the $k$th raw moment is $\mathbb{E}[X^k]$, while the $k$th central moment is $\mathbb{E}[(X-\mathbb{E}[X])^k]$.  If $k$ is even, is the $k$th central moment always bounded above the $k$th raw moment?
When $k = 2$, then $\mathbb{E}[(X-\mathbb{E}[X])^2] = \mathbb{E}[X^2]-\mathbb{E}[X]^2$, and because $\mathbb{E}[X]^2$ is always positive, it follows that this is less than or equal to $\mathbb{E}[X^2]$.  But I'm having trouble extending this to larger moments.

REPLY [2 votes]: Roberto Rastapopoulos' proof could be simplified by firstly proving the inequality holds when $a\ge 0$:
$$a^m-(a-1)^m\ge a-1;$$
Then similarly for $X\ge0$ with $E(X)=1$,
$$E(X^r)-E[(X-1)^r] = \int_{X\ge 0}[x^r-(x-1)^r]dP(\omega)\ge \int_{X\ge 0}(x-1)dP(\omega)=0.$$