Question

# This exercise finds the joint pdf of (U(1) , U(n) ), the minimum and maximum of the Ui 's. Let V1 = G1 and V2 = G2

+ · · · + Gn, so that (U(1) , U(n) ) = g(V1 , V2) = (V1 , V1 + V2). (a) What is the space of (U(1) , U(n) )? (b) The distribution of (V1 , V2) is Dirichlet. What are the parameters? Write down the pdf of (V1 , V2). (c) Find the inverse function g −1 and the absolute value of the determinant of the Jacobian of g −1 . (d) Find the pdf of (U(1) , U(n) )

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