Paper 2, Section II, F

Probability | Part IA, 2009

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

Let (X1,Y1),,(Xn,Yn)\left(X_{1}, Y_{1}\right), \ldots,\left(X_{n}, Y_{n}\right) be nn independent random points of the unit square S={(x,y):0x,y1}\mathcal{S}=\{(x, y): 0 \leqslant x, y \leqslant 1\}. We say that (Xi,Yi)\left(X_{i}, Y_{i}\right) is a maximal external point if, for each j=1,,nj=1, \ldots, n, either XjXiX_{j} \leqslant X_{i} or YjYiY_{j} \leqslant Y_{i}. (For example, in the figure below there are three maximal external points.) Determine the expected number of maximal external points.

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