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Consider n independent variables, {X1, X2, . . . , Xn} uniformly distributed over the unit...

Consider n independent variables, {X1, X2, . . . , Xn} uniformly distributed over the unit interval, (0, 1). Introduce two new random variables, M = max (X1, X2, . . . , Xn) and N = min (X1, X2, . . . , Xn).

(A) Find the joint distribution of a pair (M, N).

(B) Derive the CDF and density for M.

(C) Derive the CDF and density for N.

(D) Find moments of first and second order for variable M.

(E) Find expectation and variance for variable N.

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