2.II.12F

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a ranking of the yearly rainfalls in Cambridge over the next $n$ years: assume $a_{1}, a_{2}, \ldots, a_{n}$ is a random permutation of $1,2, \ldots, n$. Year $k$ is called a record year if $a_{i}>a_{k}$ for all $i<k$ (thus the first year is always a record year). Let $Y_{i}=1$ if year $i$ is a record year and 0 otherwise.

Find the distribution of $Y_{i}$ and show that $Y_{1}, Y_{2}, \ldots, Y_{n}$ are independent and calculate the mean and variance of the number of record years in the next $n$ years.

Find the probability that the second record year occurs at year $i$. What is the expected number of years until the second record year occurs?

*Typos? Please submit corrections to this page on GitHub.*