In your second definition for M you have the expression

((Exp[Total[
       Log[1 - (1 - (1 + 2*\[Xi]*y)*Exp[-2*\[Xi]*y])^\[Mu]]^(\[Nu] - 
        1)]] /. (\[Mu] -> RAND1)) /. (\[Nu] -> RAND2)) /. \[Xi] -> 
  RAND3

which is a tensor of rank 3. We can describe it as a 10 by 10 matrix whose entries are vectors of length 1000. Then you call Mean on {this thing}, which gives back this thing. You multiply this tensor element-by-element by a numerical vector of length 1000. I wonder if this is what you want.

n is greater than 1. I just want to check the result so put n=1

y is a list {someRandomValue}. Did you mean to do that? Once you make it a scalar (Real), M is a list of 0+i 0 so I think there's something wrong in your equation there