This is the story of the averages and variance

                          of the random variable, x[1]

                           under the constraint Ax=b

                                 for the matrix

                          [1    0    1    0    1    1]
                          [                          ]
                          [0    1    1    1    0    1]

            with the Poisson parameters given by, [1, 1, 1, 1, 1, 1]

                      for b from , [3, 3], to , [200, 200]

                          The sequence of averages is

[0.7674418605, 0.9382716049, 1.091794159, 1.232450150, 1.363042909, 1.485479789,

    1.601129684, 1.711015199, 1.815924491, 1.916480451, 2.013185609,

    2.106452441, 2.196624501, 2.283991573, 2.368800775, 2.451264882,

    2.531568668, 2.609873810, 2.686322749, 2.761041748, 2.834143358,

    2.905728416, 2.975887688, 3.044703222, 3.112249481, 3.178594287,

    3.243799626, 3.307922325, 3.371014634, 3.433124732, 3.494297149,

    3.554573150, 3.613991057, 3.672586534, 3.730392839, 3.787441049,

    3.843760249, 3.899377711, 3.954319048, 4.008608348, 4.062268304,

    4.115320321, 4.167784617, 4.219680312, 4.271025513, 4.321837383,

    4.372132214, 4.421925481, 4.471231905, 4.520065498, 4.568439615,

    4.616366992, 4.663859786, 4.710929615, 4.757587585, 4.803844328,

    4.849710023, 4.895194426, 4.940306896, 4.985056412, 5.029451601,

    5.073500750, 5.117211831, 5.160592511, 5.203650174, 5.246391932,

    5.288824640, 5.330954909, 5.372789117, 5.414333420, 5.455593765,

    5.496575897, 5.537285372, 5.577727562, 5.617907664, 5.657830710,

    5.697501572, 5.736924972, 5.776105483, 5.815047543, 5.853755454,

    5.892233389, 5.930485401, 5.968515424, 6.006327279, 6.043924680,

    6.081311235, 6.118490453, 6.155465747, 6.192240435, 6.228817750,

    6.265200835, 6.301392753, 6.337396487, 6.373214942, 6.408850950,

    6.444307272, 6.479586599, 6.514691557, 6.549624704, 6.584388541,

    6.618985506, 6.653417979, 6.687688286, 6.721798695, 6.755751426,

    6.789548645, 6.823192469, 6.856684970, 6.890028170, 6.923224050,

    6.956274545, 6.989181550, 7.021946916, 7.054572458, 7.087059952,

    7.119411134, 7.151627707, 7.183711337, 7.215663657, 7.247486266,

    7.279180732, 7.310748590, 7.342191346, 7.373510476, 7.404707426,

    7.435783615, 7.466740436, 7.497579254, 7.528301408, 7.558908212,

    7.589400955, 7.619780905, 7.650049304, 7.680207371, 7.710256304,

    7.740197280, 7.770031455, 7.799759962, 7.829383918, 7.858904418,

    7.888322538, 7.917639335, 7.946855851, 7.975973108, 8.004992109,

    8.033913844, 8.062739283, 8.091469383, 8.120105084, 8.148647309,

    8.177096969, 8.205454959, 8.233722158, 8.261899433, 8.289987638,

    8.317987612, 8.345900181, 8.373726158, 8.401466345, 8.429121531,

    8.456692491, 8.484179991, 8.511584785, 8.538907615, 8.566149211,

    8.593310294, 8.620391573, 8.647393749, 8.674317511, 8.701163536,

    8.727932496, 8.754625050, 8.781241849, 8.807783534, 8.834250737,

    8.860644082, 8.886964185, 8.913211650, 8.939387078, 8.965491056,

    8.991524168, 9.017486988, 9.043380082, 9.069204008, 9.094959318,

    9.120646557, 9.146266261, 9.171818960, 9.197305177, 9.222725428,

    9.248080223, 9.273370066, 9.298595452, 9.323756872, 9.348854810,

    9.373889745, 9.398862149]

                        This seems to be asymptotically

                                             b  0.5235267734
                    0.5912246679 0.9999878710  b

                                                                 1/2
            If we assume that this grows like a constant times, b

          then it is reasobale that the asymptotics of the averages is

                                             1/2
                               0.6645999159 b

                     The sequence of standard deviations is

[0.7261625581, 0.8065423687, 0.8729767672, 0.9299685758, 0.9800982477,

    1.024992636, 1.065748044, 1.103140693, 1.137741621, 1.169983955,

    1.200204523, 1.228670784, 1.255598881, 1.281166151, 1.305520011,

    1.328784419, 1.351064672, 1.372451015, 1.393021433, 1.412843795,

    1.431977565, 1.450475151, 1.468382998, 1.485742467, 1.502590572,

    1.518960570, 1.534882462, 1.550383415, 1.565488108, 1.580219039,

    1.594596770, 1.608640153, 1.622366516, 1.635791822, 1.648930819,

    1.661797152, 1.674403483, 1.686761574, 1.698882381, 1.710776121,

    1.722452343, 1.733919978, 1.745187405, 1.756262481, 1.767152600,

    1.777864713, 1.788405379, 1.798780782, 1.808996766, 1.819058862,

    1.828972300, 1.838742043, 1.848372799, 1.857869037, 1.867235008,

    1.876474756, 1.885592131, 1.894590805, 1.903474278, 1.912245894,

    1.920908845, 1.929466185, 1.937920836, 1.946275593, 1.954533137,

    1.962696036, 1.970766756, 1.978747659, 1.986641022, 1.994449025,

    2.002173767, 2.009817269, 2.017381476, 2.024868259, 2.032279422,

    2.039616705, 2.046881788, 2.054076285, 2.061201761, 2.068259728,

    2.075251639, 2.082178906, 2.089042894, 2.095844919, 2.102586258,

    2.109268148, 2.115891784, 2.122458326, 2.128968898, 2.135424589,

    2.141826457, 2.148175525, 2.154472789, 2.160719215, 2.166915743,

    2.173063281, 2.179162716, 2.185214909, 2.191220698, 2.197180894,

    2.203096291, 2.208967659, 2.214795748, 2.220581286, 2.226324988,

    2.232027545, 2.237689630, 2.243311903, 2.248895005, 2.254439559,

    2.259946178, 2.265415454, 2.270847968, 2.276244285, 2.281604962,

    2.286930533, 2.292221529, 2.297478464, 2.302701839, 2.307892147,

    2.313049867, 2.318175470, 2.323269411, 2.328332142, 2.333364100,

    2.338365717, 2.343337411, 2.348279591, 2.353192662, 2.358077015,

    2.362933038, 2.367761107, 2.372561591, 2.377334852, 2.382081246,

    2.386801117, 2.391494810, 2.396162655, 2.400804979, 2.405422104,

    2.410014345, 2.414582009, 2.419125397, 2.423644807, 2.428140529,

    2.432612851, 2.437062049, 2.441488399, 2.445892174, 2.450273635,

    2.454633043, 2.458970651, 2.463286715, 2.467581478, 2.471855181,

    2.476108063, 2.480340356, 2.484552289, 2.488744090, 2.492915978,

    2.497068173, 2.501200885, 2.505314326, 2.509408706, 2.513484224,

    2.517541082, 2.521579474, 2.525599598, 2.529601642, 2.533585793,

    2.537552235, 2.541501149, 2.545432714, 2.549347106, 2.553244495,

    2.557125055, 2.560988950, 2.564836346, 2.568667409, 2.572482291,

    2.576281155, 2.580064155, 2.583831443, 2.587583168, 2.591319482,

    2.595040528, 2.598746449, 2.602437388, 2.606113484, 2.609774875,

    2.613421697, 2.617054083, 2.620672164, 2.624276072, 2.627865933,

    2.631441874, 2.635004018, 2.638552489]

                        This seems to be asymptotically

                                             b  0.2686333844
                    0.6398576523 0.9999804797  b

                                                                (1/4)
           If we assume that this grows like a constant times, b

          then it is reasobale that the asymptotics of the averages is

                                            (1/4)
                              0.7016301435 b

                     This took, 1.891, seconds of CPU time