Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 9 / a / ii

The law of the unconscious statistician gives the improper integralIn particular $\mathbb{E}(X)=1/2$ and $\mathbb{E}(X^2)=1/3$, soFor $p=-1$, ${\int }_{\epsilon }^1{x}^{-1}dx=-\log \epsilon \to +\infty$; for $p<-1$, the power integral also tends to $+\infty$. Thus $\mathbb{E}(X^p)=+\infty$ for $p\le -1$, so there is no finite expected value. Negative powers need only be defined almost surely on $X>0$; the null event $X=0$ has no effect on this conclusion.