What is the expected value of the absolute difference between 2 N faced dice? What about the difference between 2 dice one with N faces and one with M faces?

While finding the expected value of 2 random variable sums or differences are simple enough, how do you deal with absolute value of differences?


  • 1
    $\begingroup$ Have you tried listing all the possibilities in some simple cases? $\endgroup$ May 20, 2012 at 20:50

1 Answer 1


Split into cases.

For the first, $$\sum_{i=1}^n \sum_{j=1}^n \dfrac{|i-j|}{n^2} = 2 \sum_{i=1}^n \sum_{j=1}^i \frac{i-j}{n^2} = \frac{n^2-1}{3n}$$

For the second, if $n < m$, $$ \sum_{i=1}^n \sum_{j=1}^m \dfrac{|i-j|}{nm} = \sum_{i=1}^n \sum_{j=1}^i \frac{i-j}{nm} + \sum_{i=1}^{n} \sum_{j=i}^m \frac{j-i}{nm} = \frac{2 n^2 - 3 n m + 3 m^2 - 2}{6m}$$

  • $\begingroup$ Is it surprising that the second formula is so unsymmetric in $m$ and $n$, with $m$ but not $n$ in the denominator? $\endgroup$ May 21, 2012 at 3:54
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    $\begingroup$ Maybe it would be if I could figure out what $n=0$ would mean. $\endgroup$ May 21, 2012 at 5:31

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