Paper 2, Section II, F

Probability | Part IA, 2014

For any function g:RRg: \mathbb{R} \rightarrow \mathbb{R} and random variables X,YX, Y, the "tower property" of conditional expectations is

E[g(X)]=E[E[g(X)Y]].E[g(X)]=E[E[g(X) \mid Y]] .

Provide a proof of this property when both X,YX, Y are discrete.

Let U1,U2,U_{1}, U_{2}, \ldots be a sequence of independent uniform U(0,1)U(0,1)-random variables. For x[0,1]x \in[0,1] find the expected number of UiU_{i} 's needed such that their sum exceeds xx, that is, find E[N(x)]E[N(x)] where

N(x)=min{n:i=1nUi>x}N(x)=\min \left\{n: \sum_{i=1}^{n} U_{i}>x\right\}

[Hint: Write E[N(x)]=E[E[N(x)U1]].]\left.E[N(x)]=E\left[E\left[N(x) \mid U_{1}\right]\right] .\right]

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