There are 100 seats on a plane. If each of the 100 passengers randomly take a seat, how many people will have his/her own seat on average?

There is a classic way to solve this problem, which is to decompose a (complicated) random variable into a sum of simple …

Suppose there are \(n\) distinct numbers \(x_1,\ldots, x_n\), and \(y_1, \ldots, y_n\) is a random permutations of them. If \(\exists k\) such that \(y_k<y_i, \forall 1\le i<k\), then we say that \(y_k\) is a record (we always count \(y_1\) as a record). What is the expected …