Probability | Poisson Distribution | Siblings Distribution


Question

Consider a large population of families, and suppose that the number of children in the different families is independent Poisson random variables with mean \(\lambda\). Show that the number of siblings of a randomly chosen child is also Poisson distributed with mean \(\lambda\).


Solution | Analytical

Let us tackle this question systematically


Solution | Simulation

There are two bar charts below




Module[{families = RandomVariate[PoissonDistribution[10], 100000],
  familiesPlot, siblings, siblingsPlot, plot},

 siblings = Select[families, # > 0 &] - 1;

 familiesPlot = Labeled[BarChart[KeySort[Counts[families]],
    LabelingFunction -> (Placed[Rotate[#, 90 °], Above] &),
    ImageSize -> 500, Frame -> True,
    FrameTicks -> {{None, None}, {Range[25], None}},
    ChartStyle -> LightBlue,
    GridLines -> {Range[25], Range[0, 16000, 1250]},
    PlotRange -> {{0, 25.5}, {0, 16000}}, AspectRatio -> 0.5],
   Rotate[Style["Counts(siblings/child)", 12], 90 °], Right];

 siblingsPlot = Labeled[BarChart[-KeySort[Counts[siblings]],
    LabelingFunction -> (Placed[Rotate[-#, 90 °], Below] &),
    ImageSize -> 500, Frame -> True, ChartStyle -> LightGreen,
    GridLines -> {Range[25], Range[0, -16000, -1250]},
    FrameTicks -> {{None, None}, {None, Range[25]}},
    PlotRange -> {{0, 25.5}, {0, -16000}}, AspectRatio -> 0.5],
   Rotate[Style["Counts(siblings/child)", 12], 90 °], Right];

 plot = Column[{familiesPlot, siblingsPlot}, Spacings -> 0];

 Export[StringReplace[NotebookFileName[], ".nb" -> ".svg"], plot]]
]