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Math Help - Geometric mean of IIDs with uniform distribution [0,1]

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    Geometric mean of IIDs with uniform distribution [0,1]

    Let X_1,X_2,\dots,X_n be n independent variables, all with the uniform distribution U(0,1). Am I correct in concluding that the geometric mean G=\sqrt[n]{X_1\cdot{X_2}\cdots{X_n}} of these variables has probability density function f_G(g) given by \frac{n^n}{(n-1)!}\left(-g\ln{g}\right)^{n-1} when 0\le{g}\le1, and zero for g outside that range? I used the uniform product distribution, along with the "change of variable" associated with the nth root function to find this.

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    Yes, it's correct
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