Probability distribution in $L_p$

Question

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.

Can anyone help me, please?

Answer 1

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 $a>p$.