Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 10F (Probability)

The event $Y_n\le z$ occurs precisely when every one of the first $n$ levels is at most $z$. Their independence gives the cumulative distribution functionFor any nonnegative random variable, the pointwise identity $Y_n={\int }_0^{\infty }{1}_{\{Y_n>z\}}dz$ and the Tonelli theorem allow interchange of expectation and integral. HenceThis identity holds as an extended nonnegative integral; finiteness of the mean is not guaranteed by continuity of $F$.

The future levels are independent of the observed maximum. Given $Y_n=z$, the first $k-1$ future levels must be at most $z$, and the $k$th must exceed $z$. Each non-overflow has probability $F(z)$ and an overflow has probability $1-F(z)$. ThereforeThis is a geometric distribution of parameter $1-F(z)$ when $F(z)<1$. Since $Y_n$ has a continuous distribution, conditioning on an exact value is understood via a regular conditional distribution, rather than division by $P(Y_n=z)=0$. The formula specifies its usual version. If $F(z)=1$ there is no possible overflow at that threshold, but such thresholds have probability zero for the random observed maximum.

The continuity of the cumulative distribution function gives the probability integral transform: $F(X_i)$ is uniform on $[0,1]$. Strict monotonicity or a probability density for $F$ is not required. Since $F$ is nondecreasing,By independence, $P(U\le u)=u^n$ for $0\le u\le 1$, so $U$ has probability density function $n{u}^{n-1}$. Average the conditional probability in part (b) using this distribution:Thus $\boxed{P(\tau =k)=n/[(k+n-1)(k+n)]}$. The expression telescopes to total mass one, so the waiting time is finite almost surely.

For the waiting time for a new record after a fixed sample, given $U=F(Y_n)$, no overflow in the first $k$ subsequent years has probability $U^k$. Averaging as in part (c) yieldsThe tail-sum formula for expectation therefore givesThe answer is $\boxed{E\tau =\infty }$. This does not contradict almost-sure finiteness: the tail probability tends to zero, but too slowly for its sum to converge.