Poissonizationとは 意味・読み方・使い方

語源

Poisson +‎ -ization.

名詞

Poissonization (usually uncountable, plural Poissonizations)

1. (mathematics, combinatorics) Analysis of a multivariate combinatorial distribution by turning interdependent variables into independent Poisson distributions.

Examples

• When randomly allocating ${\displaystyle n}$ balls into ${\displaystyle m}$ urns, each urn has an expected ${\displaystyle \lambda ={\frac {n}{m}}}$ balls. As ${\displaystyle n\to \infty }$, discrete rounds are 'blurred' into continuous time and the distribution of in an individual bin converges to a Poisson distribution, while as ${\displaystyle m\to \infty }$, covariance between two urns' counts is negligible except in extreme cases whose probability is asymptotically negligible.

• By Poissonization, the chance of all bins being filled (the coupon collector's problem) is ${\displaystyle (1-e^{-\lambda })^{m}}$; if ${\displaystyle n=m(\log m+c)}$, this becomes ${\displaystyle P=\left(1-{\frac {e^{-c}}{m}}\right)^{m}{\underset {m\to \infty }{\sim }}e^{-e^{-c}}}$, so to have a ${\displaystyle P}$ chance of filling them all (with ${\displaystyle P}$ close to ${\displaystyle 1}$), the expected number of balls is ${\displaystyle n\sim m\left(\log m-\log \log {\frac {1}{P}}\right)}$.

• When considering a single urn among the set, Poissonisation becomes an exact method, such as in this math.se question. Some authors use the term to refer specifically to this.