# 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)

<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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