Question: Let X1, ...,Xn be a random sample selected from the U[0, 1] distribution. (i) Find the distribution of variable W = Xi:n Xi1:n, 1

Let X1, ...,Xn be a random sample selected from the U[0, 1] distribution. (i) Find the distribution of variable W = Xi:n − Xi−1:n, 1 < i ≤ n. (ii) Show that the distribution of Xj:n is BETA(j,n − j + 1) for j = 1, ...,n. (ii) Find the distribution of a sample median Xk+1:n, and obtain its variance if n = 3, 7, 15, 2k + 1. Do you see any trend?

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