Notice that P(Ymin>a)=(1-a/θ)^n (which is the key point), so P(Ymin=a)=-(n/θ)*(1-a/θ)^(n-1)。 Then we use the formula: Var(X)=E(X^2)-(EX)^2 to compute Var(θ2)。
E(X^2)/(n+1)^2=the integral of -(n/θ)*(1-a/θ)^(n-1)*a^2 da from 0 to θ =the integral of -(n/θ)*(1-a/θ)^(n-1)*[(θ-a)^2+2θa-θ^2] da from 0 to θ =-θ^2+2θ*θ/(n+1)+nθ^2/(n+2) =2θ^2/(n+1)(n+2) So Var(θ2)=(n+1)*2θ^2/(n+2)-θ^2=nθ^2/(n+2)。
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