Code forhttp://math.stackexchange.com/questions/1243365

dist = EmpiricalDistribution[Array[pr, 3^2] -> Tuples[Range[3], 2]];

{inst} = FindInstance[
  sol = Reduce[
    ForAll[x,
      CDF[MarginalDistribution[dist, 1], x] ==
       CDF[MarginalDistribution[dist, 2], x]] &&
     DistributionParameterAssumptions[dist] &&
     Array[0 <= pr[#] <= 1 &, 3^2, 1, And] &&
     Expectation[x/(x + y), {x, y} \[Distributed] dist] !=
      Expectation[y/(x + y), {x, y} \[Distributed] dist], Reals],
  Array[pr, 3^2]]

PiecewiseExpand[PDF[counterExampleDist = dist /. inst, {x, y}]]

{CDF[MarginalDistribution[counterExampleDist, 1], x],
 CDF[MarginalDistribution[counterExampleDist, 2], x]}

{Expectation[x/(x + y), {x, y} \[Distributed] counterExampleDist],
 Expectation[y/(x + y), {x, y} \[Distributed] counterExampleDist]}

DiscretePlot3D[
 PDF[counterExampleDist = dist /. inst, {x, y}], {x, 1, 3}, {y, 1, 3},
  ExtentSize -> 0.9, PlotTheme -> "Web"]