Paper 2, Section II, F

Probability | Part IA, 2007

Let NN be a non-negative integer-valued random variable with

P{N=r}=pr,r=0,1,2,P\{N=r\}=p_{r}, \quad r=0,1,2, \ldots

Define EN\mathbb{E} N, and show that

EN=n=1P{Nn}.\mathbb{E} N=\sum_{n=1}^{\infty} P\{N \geqslant n\} .

Let X1,X2,X_{1}, X_{2}, \ldots be a sequence of independent and identically distributed continuous random variables. Let the random variable NN mark the point at which the sequence stops decreasing: that is, N2N \geqslant 2 is such that

X1X2XN1<XN,X_{1} \geqslant X_{2} \geqslant \ldots \geqslant X_{N-1}<X_{N},

where, if there is no such finite value of NN, we set N=N=\infty. Compute P{N=r}P\{N=r\}, and show that P{N=}=0P\{N=\infty\}=0. Determine EN\mathbb{E} N.

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