英和和英

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.