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I would like to find the distribution (CDF or PDF) of a random variable Y, which is written as

Y=X_1*X_2*...X_N/(X_1+X_2+...X_N)^N.

X_1, X_2,...X_N are N i.i.d. random variables and we know they have the same PDF f_X(x).

I know this can be solved by change of variables technique and use multi-dimensional integration. But is there a easier way to deal with it?

Thanks.