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Visualizing the Poisson Limit of the Binomial Distribution

Illustrate how a binomial distribution converges to Poisson form in the low-probability limit

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Gary Kasparov, a chess champion, plays in a tournament simultaneously against 100 amateurs. It has been estimated that he loses about 1% of such games.

Compute the probabilities of him losing 0, 2, 5, or 10 games using BinomialDistribution:

In[1]:=

Table[Probability[xk,xBinomialDistribution[100,0.01]],{k,{0,2,5,10}}]

Out[1]=

{0.366032,0.184865,0.00289779,7.00604×

-8

}

Use a Poisson approximation to compute the same probabilities:

In[2]:=

Table[Probability[xk,xPoissonDistribution[100*0.01]],{k,{0,2,5,10}}]

Out[2]=

{0.367879,0.18394,0.00306566,1.01378×

-7

}

Perform the same computation when he is playing 5 games, but with stronger opposition so that his loss probability is 20% instead:

In[3]:=

Table[Probability[xk,xBinomialDistribution[5,0.20]],{k,{0,2,5,10}}]

Out[3]=

{0.32768,0.2048,0.00032,0}

In this case the Poisson approximation is less accurate:

In[4]:=

Table[Probability[xk,xPoissonDistribution[5*0.2]],{k,{0,2,5,10}}]

Out[4]=

{0.367879,0.18394,0.00306566,1.01378×

-7

}

When the number of games is small (n = 5, p = 0.2), the blue curve deviates noticeably from the Poisson model. As the number of games increases (n = 100, p = 0.01), the orange curve almost overlaps with the Poisson curve, showing that the approximation becomes accurate for large n and small p:

In[5]:=

k=Range[0,10];binomSmall=PDF[BinomialDistribution[5,0.2],k];binomLarge=PDF[BinomialDistribution[100,0.01],k];pois=PDF[PoissonDistribution[1],k]

Out[5]=





2

6

24

120

720

5040

40320

362880

3628800



In[6]:=

ListLinePlot[{binomSmall,binomLarge,pois},PlotStyle{{Thick,Blue},{Thick,Orange},{Thick,Dashed,Darker@Green}},PlotLegends{"Binomial(5, 0.2)","Binomial(100, 0.01)","Poisson(1)"},FrameTrue,FrameLabel{"Number of losses","Probability"},LabelStyle14,ImageSizeLarge]

Out[6]=

	Binomial(5, 0.2)
	Binomial(100, 0.01)
	Poisson(1)

Related Symbols

BinomialDistribution
PoissonDistribution
BernoulliDistribution
PDF
Mean
RandomVariate
ListLinePlot

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Contributed by: Wolfram Staff