Poisson approximation to Binomial distribution 이항분포의 포아송 근사

Poisson approximation to the Binomial distribution

 

The Binomial distribution consists of two-parameter which is n and p.

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n: the number of independent trials

p: the probability of getting success.

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When 'n' in Bin(n,p) is large enough and p is close to 0, binomial distribution converges to poisson distribution.

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Binomial Distribution

 

when X follows binomial distribution, X~Bin(n,p)

X: the number of success

So binomial distribution is discrete distribution because it deals with the discrete numbers. (number of success should be like 0,1,2,3,..)

 

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Poisson Distribution

 

When X has poisson distribution,X~POI(λ)

X  : the number of event occurrences in a fixed time period.

So poisson distribution is discrete distribution because random variables should be discrete numbers.

 

 

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Conditions for Poisson Approximation to Binomial

 

 

n is large enough (n> 100)

p is very close to 0 (p< 0.01)

 

 

Then Bin(n,p) is approximately the same with POI(np)

That is, np can approximate λ

 

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Example of Poisson Approximation to Binomial

Poisson approximation to Binomial is useful when n is large enough and p is very close to 0 because it is conflict to compute binomial distribution.

For example, when X~BIN(100000, 0.00007)

It seems conflict to find P(X=x).

For this case, we use Poisso Approximation.

 

 

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Graph(Using R) - Poisson Approximation

 

graph using R

You can see that when n is large enough and p is close to 0, Bin(n,p) graph looks similar to POI(np)