Let \(Y_{1}, \ldots, Y_{n}\) be independent and identically distributed standard exponential variables, and let \(0 \leq Y_{(1)} \leq Y_{(2)} \leq \cdots \leq Y_{(n)}\) be the order statistics. Show that \(Y_{(1)} \geq t\) if and only if \(Y_{i} \geq t\) for \(1 \leq i \leq n\), and deduce that \(n Y_{(1)}\) is exponentially distributed with unit mean. Hence or otherwise, show that the increments \((n-r)\left(Y_{(r+1)}-Y_{(r)}ight)\) are independent and identically distributed. Find the mean and variance of the maximum \(Y_{(n)}\), with asymptotic values for large \(n\).