bugs - Strange behavior of RandomVariate and PascalDistribution

Observe:

sr = 666;

d = PascalDistribution[1, 1/50];
od = OrderDistribution[{d, 10}, 1];

Column[{Mean[od] // N,
  SeedRandom[sr];
  Dimensions@(t1 = RandomVariate[d, {1000, 10}]),
  SeedRandom[sr];

  Dimensions@(t2 = Table[RandomVariate[d, 10], 1000]),
  SeedRandom[sr];
  Dimensions@(t3 = Partition[RandomVariate[d, 10000], 10]),
  SeedRandom[sr];
  Dimensions@(t4 = 
     ArrayReshape[Table[RandomVariate[d, 100], 100], {1000, 10}]),
  Min /@ t1 // Mean // N,
  Min /@ t2 // Mean // N,
  Min /@ t3 // Mean // N,
  Min /@ t4 // Mean // N}]

5.46666

{1000,10}

{1000,10}

{1000,10}

{1000,10}

5.463

6.484

5.463

5.463

The results from Min/@t...//Mean//N should be the same for all four cases.

Note the result (using the 666 seed) of 6.484 for the t2 case. This is consistently wacky (~+1 from actual expectation shown by the mean of the order distribution).

Changing the inner cardinality for the t4 case to anything 80 or above (e.g. ArrayReshape[Table[RandomVariate[d, 80], 125], {1000, 10}]keeps it consistent, but lowering it to say 50 in the RV generation results in also off-kilter results.

I'd venture there is some kind of heuristic switching of methods going on based on requested number of samples, and perhaps a bug in the low-count algorithm.

On 10.3 Windows, same results on 9.X Windows.

I'd appreciate verification (and explanation if I've pulled a DOH and there's a reason for this behavior).

Answer

Just an extended comment: Rather than a consistent shift for the values in the minimum values for t2, it appears that the distribution is completely different than for t1, t3, and t4. Here's a figure showing that:

h[x_, label_] := Histogram[Min /@ x, {1}, "PDF", PlotRange -> {{0, 30}, {0, 0.20}}, 
  PlotLabel -> Style[label, Bold, Larger]]
GraphicsGrid[{{h[t1, "t1"], h[t2, "t2"]}, {h[t3, "t3"], h[t4, "t4"]}}, ImageSize -> Large]

Comments

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