Poisson distribution

<what is Poisson distribution?>

Poisson distribution can be derived from binomial distribution.


<what is binomial distribution?>

preparing n boxes and each box can take p or q below.


(p/q)(p/q)(p/q)・・・(p/q)


all combinations of p and q are

(p+q)^n = ΣnCx p^x q^(n-x)


here, when p+q = 1, nCx p^x q^(n-x) is probability density.

let nCx p^x q^(n-x) be f(x).


f(x) is binomial distribution.

 

<properties of binomial distribution>

the mean value of f(x) is μ = np.

the variance of f(x) σ^2 = npq = np(1-p).


<Poisson distribution again>

n->∞, p->0 subject to np = μ = const.

nCx p^x q^(n-x) -> (μ^x e^(-μ))/x!

 

roughly speaking,

nCx        -> 1/x!

p^x        -> μ^x

q^(n-x)  -> e^(-μ)

 

, which would help you remember the distribution.

 

 

<example of Poisson distribution>

radioisotope polonium 210 emits alpha particles 100%.

properties of it:

decay constant λ=3.4 * 10^(-6)[/min], μ=12/min.

then, what is the probability of 15/min?


n is so many, p(λ) is so small, μ=const.

so that 210Po should obeys Poisson distribution.


f(x) = 12^x e^(-12))/x!


therefore 


f(15) = 7.24 * 10^(-2)

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