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

For the continuous uniform distribution on $[0,1]$, the probability density function isChanging the values at the endpoints does not change any probabilities, because individual points have zero probability.

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.

Use the covariance identity and the moments from (ii):The positive value is consistent with both $X$ and $X^2$ increasing together on $[0,1]$.