Mathematical expectation of the average distance of some set of pointsIn cartesian plane, N points are randomly generated within a unit square defined by points (0,0)(0,1)(1,1)(1,0), with uniform distribution. What is the mathematical expectation of the average distance of all distances between all points (distances between the point and the same point (which are obviously 0) are excluded from the calculation of the average).There are actually two versions of this question:- Distance is defined as Euclidian.- Distance is defined as Manhattan.The context of this question is some performance benchmarking in computer science, which are of practical nature and far from pure mathemathics, so I am not going the bother the reader with these details.

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hottchevymanzm · Accepted Answer

Step 1You&#039;re looking for      E        (          1                                    (                                    N          2                                      )                                      ∑              1        ⩽        i        &amp;lt;        j        ⩽        N              d    (          P      i        ,          P      j        )    )    ;Step 2By linearity of expectation, that would be      1                            (                            N        2                              )                          ∑          1      ⩽      i      &amp;lt;      j      ⩽      N            E              (        d  (      P    i    ,      P    j    )            )        .By (assumed) independence of the distribution of the points       P    i  ,       E              (        d  (      P    i    ,      P    j    )            )       is the same for all pairs   i  ≠  j.

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