91Ó°ÊÓ

The given distribution is \(f(x)=e^{-x}\) for \(0<x<\infty\), zero elsewhere. This is an exponential distribution, a continuous probability distribution of the form \(f(x|λ) = λe^{-λx}\), where the rate parameter λ is a positive real number. In this case, λ is equal to 1.

The order statistics of a random sample \(Y_{1}0\), and 0 elsewhere.

The limiting distribution of \(Z_{n}=\left(Y_{n}-\log n\right)\) can be found by applying the limit theorem. To derive this, we can write \(Z_n = Y_n - \log n\). As n approaches infinity, the term \(\log n\) grows without bound, but more slowly than \(Y_n\). Because the exponential distribution is memoryless, it implies that the distribution of the remaining life time is the same as the distribution of the life time itself. Hence, as n approaches infinity, it can be seen that the \(Z_n\) converges in distribution to a standard exponential distribution. Therefore, the limiting distribution of \(Z_{n}=\left(Y_{n}-\log n\right)\) is given by an exponential distribution with parameter 1, that is \(f(z) = e^{-z}\), for \(z > 0\), and 0 elsewhere.