Let $A$ be a random matrix of size $m\times m$ with integer entries $-n\ldots n$. Each value should have the same probability. What is the expectation of the random variable

$$X := |\det A|$$

Can the variance of $X$ also be calculated ?

  • $\begingroup$ Do you seriously think this has a simple answer? $\endgroup$ – Did Jul 26 '14 at 20:36
  • $\begingroup$ Since the expectation is a linear function, I have some hope that the problem is not too difficult. But it is clear that the expectation of absolut values is not so easy to calculate. $\endgroup$ – Peter Jul 26 '14 at 20:42
  • $\begingroup$ The second sentence of your comment should suffice to destroy the "hope" expressed in the first sentence. $\endgroup$ – Did Jul 26 '14 at 21:12
  • $\begingroup$ web.mit.edu/18.338/www/Acta05rmt.pdf $\endgroup$ – Count Iblis Jul 26 '14 at 21:20
  • 2
    $\begingroup$ @Count: That's a long document. Can you point us to the relevant part, and maybe briefly summarize what's to be found there? $\endgroup$ – Nate Eldredge Jul 26 '14 at 21:33

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