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?
Thanks
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 $n < m$, $$ \sum_{i=1}^n \sum_{j=1}^m \dfrac{|i-j|}{nm} = \sum_{i=1}^n \sum_{j=1}^i \frac{i-j}{nm} + \sum_{i=1}^{n} \sum_{j=i}^m \frac{j-i}{nm} = \frac{2 n^2 - 3 n m + 3 m^2 - 2}{6m}$$