Poisson limit theorem

Poisson limit theorem "Poisson theorem" redirects here. For the "Poisson's theorem" in Hamiltonian mechanics, see Poisson bracket § Constants of motion.

In probability theory, the law of rare events or Poisson limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions.[1] The theorem was named after Siméon Denis Poisson (1781–1840). A generalization of this theorem is Le Cam's theorem.

For broader coverage of this topic, see Poisson distribution § Law of rare events. Contents 1 Theorem 2 Proofs 2.1 Alternative proof 2.2 Ordinary generating functions 3 See also 4 References Theorem Let {displaystyle p_{n}} be a sequence of real numbers in {displaystyle [0,1]} such that the sequence {displaystyle np_{n}} converges to a finite limit {displaystyle lambda } . Then: {displaystyle lim _{nto infty }{n choose k}p_{n}^{k}(1-p_{n})^{n-k}=e^{-lambda }{frac {lambda ^{k}}{k!}}} Proofs {displaystyle {begin{aligned}lim limits _{nrightarrow infty }{n choose k}p_{n}^{k}(1-p_{n})^{n-k}&simeq lim _{nto infty }{frac {n(n-1)(n-2)dots (n-k+1)}{k!}}left({frac {lambda }{n}}right)^{k}left(1-{frac {lambda }{n}}right)^{n-k}\&=lim _{nto infty }{frac {n^{k}+Oleft(n^{k-1}right)}{k!}}{frac {lambda ^{k}}{n^{k}}}left(1-{frac {lambda }{n}}right)^{n-k}\&=lim _{nto infty }{frac {lambda ^{k}}{k!}}left(1-{frac {lambda }{n}}right)^{n-k}end{aligned}}} .

Since {displaystyle lim _{nto infty }left(1-{frac {lambda }{n}}right)^{n}=e^{-lambda }} and {displaystyle lim _{nto infty }left(1-{frac {lambda }{n}}right)^{-k}=1} This leaves {displaystyle {n choose k}p^{k}(1-p)^{n-k}simeq {frac {lambda ^{k}e^{-lambda }}{k!}}.} Alternative proof Using Stirling's approximation, we can write: {displaystyle {begin{aligned}{n choose k}p^{k}(1-p)^{n-k}&={frac {n!}{(n-k)!k!}}p^{k}(1-p)^{n-k}\&simeq {frac {{sqrt {2pi n}}left({frac {n}{e}}right)^{n}}{{sqrt {2pi left(n-kright)}}left({frac {n-k}{e}}right)^{n-k}k!}}p^{k}(1-p)^{n-k}\&={sqrt {frac {n}{n-k}}}{frac {n^{n}e^{-k}}{left(n-kright)^{n-k}k!}}p^{k}(1-p)^{n-k}.end{aligned}}} Letting {displaystyle nto infty } and {displaystyle np=lambda } : {displaystyle {begin{aligned}{n choose k}p^{k}(1-p)^{n-k}&simeq {frac {n^{n},p^{k}(1-p)^{n-k}e^{-k}}{left(n-kright)^{n-k}k!}}\&={frac {n^{n}left({frac {lambda }{n}}right)^{k}left(1-{frac {lambda }{n}}right)^{n-k}e^{-k}}{n^{n-k}left(1-{frac {k}{n}}right)^{n-k}k!}}\&={frac {lambda ^{k}left(1-{frac {lambda }{n}}right)^{n-k}e^{-k}}{left(1-{frac {k}{n}}right)^{n-k}k!}}\&simeq {frac {lambda ^{k}left(1-{frac {lambda }{n}}right)^{n}e^{-k}}{left(1-{frac {k}{n}}right)^{n}k!}}.end{aligned}}} As {displaystyle nto infty } , {displaystyle left(1-{frac {x}{n}}right)^{n}to e^{-x}} so: {displaystyle {begin{aligned}{n choose k}p^{k}(1-p)^{n-k}&simeq {frac {lambda ^{k}e^{-lambda }e^{-k}}{e^{-k}k!}}\&={frac {lambda ^{k}e^{-lambda }}{k!}}end{aligned}}} Ordinary generating functions It is also possible to demonstrate the theorem through the use of ordinary generating functions of the binomial distribution: {displaystyle G_{operatorname {bin} }(x;p,N)equiv sum _{k=0}^{N}left[{binom {N}{k}}p^{k}(1-p)^{N-k}right]x^{k}={Big [}1+(x-1)p{Big ]}^{N}} by virtue of the binomial theorem. Taking the limit {displaystyle Nrightarrow infty } while keeping the product {displaystyle pNequiv lambda } constant, we find {displaystyle lim _{Nrightarrow infty }G_{operatorname {bin} }(x;p,N)=lim _{Nrightarrow infty }left[1+{frac {lambda (x-1)}{N}}right]^{N}=mathrm {e} ^{lambda (x-1)}=sum _{k=0}^{infty }left[{frac {mathrm {e} ^{-lambda }lambda ^{k}}{k!}}right]x^{k}} which is the OGF for the Poisson distribution. (The second equality holds due to the definition of the exponential function.) See also De Moivre–Laplace theorem Le Cam's theorem References ^ Papoulis, Athanasios; Pillai, S. Unnikrishna. Probability, Random Variables, and Stochastic Processes (4th ed.). Categories: Probability theorems