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?


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 $a>p$.

  • $\begingroup$ Thank you for the answer, but why to see if it is in $L_p$ we have to compute the moments? $\endgroup$ – user149240 May 2 at 5:35

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