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

• As per the question, the number of children in different families is an independent Poisson Distribution \[P\{X=k\} = \frac{e^{-\lambda} \lambda^k}{k!} \mid 0 \le k \le \infty \]
• When we want to find the distribution of the siblings, we need families with at least 1 child
• This means that the distribution the number of families in this \[P\{Y=(m+1)\} = \frac{e^{-\lambda} \lambda^{(m+1)}}{(m+1)!} \mid 0 \le m \le \infty \] where the \(1\) from \(m + 1\) is the child for which we are studying the siblings. Which is a Poisson Distribution's probability mass function.

Solution | Simulation

There are two bar charts below

• The chart on the top shows Poisson Distribution with \(\lambda = 10\)
• The one on the bottom is derived from the above distribution by selecting those families with more than one child and subtracting \(1\) from it.

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]]
]