Paper 2, Section II, F

Probability | Part IA, 2010

The yearly levels of water in the river Camse are independent random variables X1,X2,X_{1}, X_{2}, \ldots, with a given continuous distribution function F(x)=P(Xix),x0F(x)=\mathbb{P}\left(X_{i} \leqslant x\right), x \geqslant 0 and F(0)=0F(0)=0. The levels have been observed in years 1,,n1, \ldots, n and their values X1,,XnX_{1}, \ldots, X_{n} recorded. The local council has decided to construct a dam of height

Yn=max[X1,,Xn]Y_{n}=\max \left[X_{1}, \ldots, X_{n}\right]

Let τ\tau be the subsequent time that elapses before the dam overflows:

τ=min[t1:Xn+t>Yn]\tau=\min \left[t \geqslant 1: X_{n+t}>Y_{n}\right]

(a) Find the distribution function P(Ynz),z>0\mathbb{P}\left(Y_{n} \leqslant z\right), z>0, and show that the mean value EYn=0[1F(z)n]dz.\mathbb{E} Y_{n}=\int_{0}^{\infty}\left[1-F(z)^{n}\right] \mathrm{d} z .

(b) Express the conditional probability P(τ=kYn=z)\mathbb{P}\left(\tau=k \mid Y_{n}=z\right), where k=1,2,k=1,2, \ldots and z>0z>0, in terms of FF.

(c) Show that the unconditional probability

P(τ=k)=n(k+n1)(k+n),k=1,2,\mathbb{P}(\tau=k)=\frac{n}{(k+n-1)(k+n)}, \quad k=1,2, \ldots

(d) Determine the mean value Eτ\mathbb{E} \tau.

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