# Probability distribution in $L_p$

I am stuck on this exercise.

Let $$F(x) = 1-1/x^a$$ for $$x\geq 1$$ be a distribution. For which values of $$a$$, $$F(x) \in L_p$$?

I tried to study vários integral functions, but I do not really know how to solve it.

Let $$X\sim F$$ i.e let $$X$$ have distribution function $$F$$. Then $$X\geq 0$$ with probability one. To compute the moments of $$X$$ we can use the tail formula, namely, \begin{align} EX^p=\int_0^\infty px^{p-1}P(X>x)\, dx&=\int_0^1px^{p-1}\, dx+p\int_1^\infty\frac{x^{p-1}}{x^a}\, dx\\ &=1+p\int_1^\infty \frac{1}{x^{1+a-p}}\, dx \end{align} which is finite iff $$a-p>0$$ i.e $$a>p$$.
• Thank you for the answer, but why to see if it is in $L_p$ we have to compute the moments? – user149240 May 2 at 5:35