# Proving that the absolute moment of order a exists

Given $$f(x)= \frac{ka^k}{(x+a)^{k+1}}$$ with $x \geq 0, a > 0$.

Show $E[|X|^a]< \infty$ for $a < k$.

I am not really sure what theorems can be used to prove this. What I've been thinking of using are:

$E[|X|^a]<\infty$ iff $X^{a-1} P(|X| \geq x)$ is integrable over $(0,\infty)$

or

If a random variable $X$ satisfies $n^aP(|X|>n)\to0$ as $n\to\infty$ for some $a>0$, then $E[|X|^b]<\infty$ for $0 < b < a$.

I think the more appropriate one is the second? But how do you show $n^aP(|X|>n)\to0$ as $n\to\infty$? I am quite new with stats and I don't know how to evaluate this.

Any help is very much appreciated.

• Your question is ambiguous because the variable "$a$" appears to be playing two distinct roles (for defining $f$ and as the power of $|X|$ in the expectation). Regardless, this is purely a mathematical exercise to test your understanding of improper integrals, so being new to stats is no obstacle to solving it. – whuber Sep 23 '15 at 12:59
• Nevermind I already solved this. Thanks for the answer though! I used the second one and with enough review about limits I was able to answer it. Sorry for posting so abruptlhy! – Rinrin Sep 24 '15 at 7:07