Uniform Distribution – Are Differences Between Uniformly Distributed Numbers Uniformly Distributed?: Detailed Analysis

distributionsuniform distribution

We roll a 6-sided die a large number of times.

Calculating the difference (absolute value) between a roll and its preceding roll, are the differences expected to be uniformly distributed?

To illustrate with 10 rolls:

roll num  result diff
1           1     0
2           2     1
3           1     1
4           3     2
5           3     0
6           5     2
7           1     4
8           6     5
9           4     2
10          4     0

Would the diff values be uniformly distributed?

Best Answer

No it is not uniform

You can count the $36$ equally likely possibilities for the absolute differences 

     second 1   2   3   4   5   6
first                           
1           0   1   2   3   4   5
2           1   0   1   2   3   4
3           2   1   0   1   2   3
4           3   2   1   0   1   2
5           4   3   2   1   0   1
6           5   4   3   2   1   0

which gives a probability distribution for the absolute differences of 

0    6/36  1/6
1   10/36  5/18
2    8/36  2/9
3    6/36  1/6
4    4/36  1/9
5    2/36  1/18
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