Probability | Part IA, 2005

Let a1,a2,,ana_{1}, a_{2}, \ldots, a_{n} be a ranking of the yearly rainfalls in Cambridge over the next nn years: assume a1,a2,,ana_{1}, a_{2}, \ldots, a_{n} is a random permutation of 1,2,,n1,2, \ldots, n. Year kk is called a record year if ai>aka_{i}>a_{k} for all i<ki<k (thus the first year is always a record year). Let Yi=1Y_{i}=1 if year ii is a record year and 0 otherwise.

Find the distribution of YiY_{i} and show that Y1,Y2,,YnY_{1}, Y_{2}, \ldots, Y_{n} are independent and calculate the mean and variance of the number of record years in the next nn years.

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

Typos? Please submit corrections to this page on GitHub.