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https://texercises.com/exercise/binomial-and-poisson-distribution/
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Exercise:
Show that as Nrightarrow infty and prightarrow muNp is a finite constant the binomial distribution ts to the Poisson distribution.

Solution:
The binomial distribution is defined as bn; N p fracN!N-n!n!p^n-p^N-n Since muNp rewriting the probability mass function of the binomial distribution with pfracmuN gives bn; N mu fracmu^nn!fracN!N-n!N^nleft-fracmuNright^Nleft-fracmuNright^-n Taking limits as Nrightarrow infty of each of the terms above individually: lim_Nrightarrow infty fracN!N-n!N^n lim_Nrightarrow infty fracN-n+N-n+...NN^n lim_Nrightarrow infty fracN-n+NfracN-n+N...fracNN lim_Nrightarrow infty left-fracmuNright^N e^-mu lim_Nrightarrow infty left-fracmuNright^-n as we have lim_Nrightarrow infty left+fracxright^x e Putting everything together we get lim_Nrightarrow infty bn; N mu e^-mufracmu^nn!
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Exercise:
Show that as Nrightarrow infty and prightarrow muNp is a finite constant the binomial distribution ts to the Poisson distribution.

Solution:
The binomial distribution is defined as bn; N p fracN!N-n!n!p^n-p^N-n Since muNp rewriting the probability mass function of the binomial distribution with pfracmuN gives bn; N mu fracmu^nn!fracN!N-n!N^nleft-fracmuNright^Nleft-fracmuNright^-n Taking limits as Nrightarrow infty of each of the terms above individually: lim_Nrightarrow infty fracN!N-n!N^n lim_Nrightarrow infty fracN-n+N-n+...NN^n lim_Nrightarrow infty fracN-n+NfracN-n+N...fracNN lim_Nrightarrow infty left-fracmuNright^N e^-mu lim_Nrightarrow infty left-fracmuNright^-n as we have lim_Nrightarrow infty left+fracxright^x e Putting everything together we get lim_Nrightarrow infty bn; N mu e^-mufracmu^nn!
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inpp, particle-physics, poisson
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Difficulty
(2, default)
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0 (default)
Language
ENG (English)
Type
Show
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