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

$$Y_{n}=\max \left[X_{1}, \ldots, X_{n}\right]$$

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

$$\tau=\min \left[t \geqslant 1: X_{n+t}>Y_{n}\right]$$

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

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

(c) Show that the unconditional probability

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

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