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Random Variableshard
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Let X1,X2,…,XnX_1, X_2, \ldots, X_nX1​,X2​,…,Xn​ be i.i.d. from a continuous distribution with CDF FFF. Let X(1)≤X(2)≤⋯≤X(n)X_{(1)} \le X_{(2)} \le \cdots \le X_{(n)}X(1)​≤X(2)​≤⋯≤X(n)​ be the order statistics. What is the distribution of Uk=F(X(k))U_k = F(X_{(k)})Uk​=F(X(k)​)?