Let X1, . . . , Xn be a sample from cdf F and denote the order statistics by X(1), X(2),...

Let X₁, . . . , X_n be a sample from cdf F and denote the order statistics by X₍₁₎, X₍₂₎, . . ., X_(n). We will assume that F is continuous, with density function f . From Theorem A in Section 3.7, the density function of X(k) is

a. Find the mean and variance of X_(k) from a uniform distribution on [0, 1]. You will need to use the fact that the density of X_(k) integrates to 1. Show that

b. Find the approximate mean and variance of Y_(k), the kth-order statistic of a sample of size n from F. To do this, let