Let $X$ and $Y$ be two independent uniformly distributed random variables on $[0,1]$. Prove that $\mathbb{E} X^{k}=\frac{1}{k+1}$ and $\mathbb{E}(X Y)^{k}=\frac{1}{(k+1)^{2}}$, and find $\mathbb{E}(1-X Y)^{k}$, where $k$ is a non-negative integer.

Let $\left(X_{1}, Y_{1}\right), \ldots,\left(X_{n}, Y_{n}\right)$ be $n$ independent random points of the unit square $\mathcal{S}=\{(x, y): 0 \leqslant x, y \leqslant 1\}$. We say that $\left(X_{i}, Y_{i}\right)$ is a maximal external point if, for each $j=1, \ldots, n$, either $X_{j} \leqslant X_{i}$ or $Y_{j} \leqslant Y_{i}$. (For example, in the figure below there are three maximal external points.) Determine the expected number of maximal external points.