Sheldon Ross 10: Example 2.10

Question Suppose that the number of typographical errors on a single page of this book has a Poisson distribution with parameter \(\lambda = 1\). Calculate the probability that there is at least Sone error on this page.
This is a problem that has a straightforward calculation. In addition to solving the problem, we will also do a simulation to get a realization of the mean reaching \(\lambda\) in the long run. Also trying some cool visualizations here 😉

Step 01: Numerical Solution

We need the probability that the number of errors is at-least one. \[P\{X \ge 1\} = 1 - P\{X = 0\} = e^{-\lambda} \frac{\lambda^{x}}{x!}\]

Step 02: Simulation and Visualization

In each of the squares, the thin horizontal lines are the number of events that happened in that second. The red line tracks mean as the events accumulate. See how the mean reaches the equilibrium gradually (even though they start out in a rugged manner).

    plot[] := Module[{poisson, sample = 500, movingProbability, scale = 0.001},
    poisson = RandomVariate[PoissonDistribution[1], sample];
    movingProbability = Transpose[{Reverse[Accumulate[poisson]/Range[sample]], scale*Range[sample]}];
    Graphics[{
        Table[{Opacity[0.1], Line[{{0, scale*r}, {poisson[[r]], scale r}}]}, {r, 1, sample}], {Red, Line[movingProbability]}},
        ImageSize -> 240,
        AspectRatio -> 1,
        Frame -> True,
        FrameTicks -> {Automatic, None},
        PlotRange -> {{0, 6}, All}]
    ]
    plot[number_] := Parallelize[Table[plot[], number]]
    Export[StringReplace[NotebookFileName[], ".nb" -> ".svg"], GraphicsGrid[Partition[plot[30], 3]], ImageSize -> 754, ImageResolution -> 600]