Poisson distribution

The distribution was first introduced by Siméon Denis Poisson (1781–1840) and published together with his probability theory in his work Recherches sur la probabilité des jugements en matière criminelle et en matière civile (1837).^{[4]: 205-207} The work theorized about the number of wrongful convictions in a given country by focusing on certain random variables N that count, among other things, the number of discrete occurrences (sometimes called "events" or "arrivals") that take place during a time-interval of given length. The result had already been given in 1711 by Abraham de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus .^{[5]: 219 [6]: 14-15 [7]: 193 [8]: 157} This makes it an example of Stigler's law and it has prompted some authors to argue that the Poisson distribution should bear the name of de Moivre.^[9][10]

In 1860, Simon Newcomb fitted the Poisson distribution to the number of stars found in a unit of space.^[11] A further practical application was made by Ladislaus Bortkiewicz in 1898. Bortkiewicz showed that the frequency with which soldiers in the Prussian army were accidentally killed by horse kicks could be well modeled by a Poisson distribution.^{[12]: 23-25}.

A discrete random variable X is said to have a Poisson distribution with parameter ${\displaystyle \lambda >0}$ if it has a probability mass function given by:^[2]: 60 $${\displaystyle f(k;\lambda )=\Pr(X{=}k)={\frac {\lambda ^{k}e^{-\lambda }}{k!}},}$$ where

• k is the number of occurrences (${\displaystyle k=0,1,2,\ldots }$)
• e is Euler's number (${\displaystyle e=2.71828\ldots }$)
• k! = k(k–1) ··· (3)(2)(1) is the factorial.

The positive real number λ is equal to the expected value of X and also to its variance.^[13] $${\displaystyle \lambda =\operatorname {E} (X)=\operatorname {Var} (X).}$$

The Poisson distribution can be applied to systems with a large number of possible events, each of which is rare. The number of such events that occur during a fixed time interval is, under the right circumstances, a random number with a Poisson distribution.

The equation can be adapted if, instead of the average number of events ${\displaystyle \lambda ,}$ we are given the average rate ${\displaystyle r}$ at which events occur. Then ${\displaystyle \lambda =rt,}$ and:^[14] $${\displaystyle P(k{\text{ events in interval }}t)={\frac {(rt)^{k}e^{-rt}}{k!}}.}$$

The Poisson distribution may be useful to model events such as:

• the number of meteorites greater than 1-meter diameter that strike Earth in a year;
• the number of laser photons hitting a detector in a particular time interval;
• the number of students achieving a low and high mark in an exam; and
• locations of defects and dislocations in materials.

Examples of the occurrence of random points in space are: the locations of asteroid impacts with earth (2-dimensional), the locations of imperfections in a material (3-dimensional), and the locations of trees in a forest (2-dimensional).^[15]

The Poisson distribution is an appropriate model if the following assumptions are true:

• k, a nonnegative integer, is the number of times an event occurs in an interval.
• The occurrence of one event does not affect the probability of a second event.
• The average rate at which events occur is independent of any occurrences.
• Two events cannot occur at exactly the same instant.

If these conditions are true, then k is a Poisson random variable; the distribution of k is a Poisson distribution.

The Poisson distribution is also the limit of a binomial distribution, for which the probability of success for each trial is ${\displaystyle p={\frac {\lambda }{n}}}$, where ${\displaystyle \lambda }$ is the expectation and ${\displaystyle n}$ is the number of trials, in the limit that ${\displaystyle n\to \infty }$ with ${\displaystyle \lambda }$ kept constant ^[16][17] (see Related distributions): $${\displaystyle \lim _{n\to \infty }{\dbinom {n}{k}}\left({\frac {\lambda }{n}}\right)^{k}\,\left(1-{\frac {\lambda }{n}}\right)^{n-k}={\frac {\lambda ^{k}}{k!}}\,e^{-\lambda }}$$ The Poisson distribution may also be derived from the differential equations^[18][19][20] $${\displaystyle {\frac {d\,P_{k}(t)}{dt}}=\lambda \,{\Big (}P_{k-1}(t)-P_{k}(t){\Big )}}$$ with initial conditions ${\displaystyle P_{k}(0)=\delta _{k0}}$ and evaluated at ${\displaystyle t=1}$

The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups). The non-constant arrival rate may be modeled as a mixed Poisson distribution, and the arrival of groups rather than individual students as a compound Poisson process.

The number of magnitude 5 earthquakes per year in a country may not follow a Poisson distribution, if one large earthquake increases the probability of aftershocks of similar magnitude.

Examples in which at least one event is guaranteed are not Poisson distributed; but may be modeled using a zero-truncated Poisson distribution.

Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a zero-inflated model.

• The expected value of a Poisson random variable is λ.
• The variance of a Poisson random variable is also λ.
• The coefficient of variation is ${\textstyle \lambda ^{-1/2},}$ while the index of dispersion is 1.^[8]: 163
• The mean absolute deviation about the mean is^[8]: 163 $${\displaystyle \operatorname {E} [\ |X-\lambda |\ ]={\frac {2\lambda ^{\lfloor \lambda \rfloor +1}e^{-\lambda }}{\lfloor \lambda \rfloor !}}.}$$
• The mode of a Poisson-distributed random variable with non-integer λ is equal to ${\displaystyle \lfloor \lambda \rfloor ,}$ which is the largest integer less than or equal to λ. This is also written as floor(λ). When λ is a positive integer, the modes are λ and λ − 1.
• All of the cumulants of the Poisson distribution are equal to the expected value λ. The n-th factorial moment of the Poisson distribution is λⁿ.
• The expected value of a Poisson process is sometimes decomposed into the product of intensity and exposure (or more generally expressed as the integral of an "intensity function" over time or space, sometimes described as "exposure").^[22]

Bounds for the median (${\displaystyle \nu }$) of the distribution are known and are sharp:^[23] $${\displaystyle \lambda -\ln 2\leq \nu <\lambda +{\frac {1}{3}}.}$$

The higher non-centered moments m_k of the Poisson distribution are Touchard polynomials in λ: $${\displaystyle m_{k}=\sum _{i=0}^{k}\lambda ^{i}{\begin{Bmatrix}k\\i\end{Bmatrix}},}$$ where the braces { } denote Stirling numbers of the second kind.^[24][1]: 6 In other words, $${\displaystyle E[X]=\lambda ,\quad E[X(X-1)]=\lambda ^{2},\quad E[X(X-1)(X-2)]=\lambda ^{3},\cdots }$$ When the expected value is set to λ = 1, Dobinski's formula implies that the n‑th moment is equal to the number of partitions of a set of size n.

A simple upper bound is:^[25] $${\displaystyle m_{k}=E[X^{k}]\leq \left({\frac {k}{\log(k/\lambda +1)}}\right)^{k}\leq \lambda ^{k}\exp \left({\frac {k^{2}}{2\lambda }}\right).}$$

If ${\displaystyle X_{i}\sim \operatorname {Pois} (\lambda _{i})}$ for ${\displaystyle i=1,\dotsc ,n}$ are independent, then ${\textstyle \sum _{i=1}^{n}X_{i}\sim \operatorname {Pois} \left(\sum _{i=1}^{n}\lambda _{i}\right).}$^[26]: 65 A converse is Raikov's theorem, which says that if the sum of two independent random variables is Poisson-distributed, then so are each of those two independent random variables.^[27][28]

It is a maximum-entropy distribution among the set of generalized binomial distributions ${\displaystyle B_{n}(\lambda )}$ with mean ${\displaystyle \lambda }$ and ${\displaystyle n\to \infty }$,^[29] where a generalized binomial distribution is defined as a distribution of the sum of N independent but not identically distributed Bernoulli variables.

• The Poisson distributions are infinitely divisible probability distributions.^{[30]: 233 [8]: 164}
• The directed Kullback–Leibler divergence of ${\displaystyle P=\operatorname {Pois} (\lambda )}$ from ${\displaystyle P_{0}=\operatorname {Pois} (\lambda _{0})}$ is given by$${\displaystyle \operatorname {D} _{\text{KL}}(P\parallel P_{0})=\lambda _{0}-\lambda +\lambda \log {\frac {\lambda }{\lambda _{0}}}.}$$
• If ${\displaystyle \lambda \geq 1}$ is an integer, then ${\displaystyle Y\sim \operatorname {Pois} (\lambda )}$ satisfies ${\displaystyle \Pr(Y\geq E[Y])\geq {\frac {1}{2}}}$ and ${\displaystyle \Pr(Y\leq E[Y])\geq {\frac {1}{2}}.}$^{[31][failed verification – see discussion]}
• Bounds for the tail probabilities of a Poisson random variable ${\displaystyle X\sim \operatorname {Pois} (\lambda )}$ can be derived using a Chernoff bound argument.^{[32]: 97-98} $${\displaystyle {\begin{aligned}P(X\geq x)&\leq {\frac {\left(e\lambda \right)^{x}e^{-\lambda }}{x^{x}}},&{\text{ for }}x>\lambda ,\\[1ex]P(X\leq x)&\leq {\frac {\left(e\lambda \right)^{x}e^{-\lambda }}{x^{x}}},&{\text{ for }}x<\lambda .\end{aligned}}}$$
• The upper tail probability can be tightened (by a factor of at least two) as follows:^[33]$${\displaystyle P(X\geq x)\leq {\frac {e^{-\operatorname {D} _{\text{KL}}(Q\parallel P)}}{\max {(2,{\sqrt {4\pi \operatorname {D} _{\text{KL}}(Q\parallel P)}}})}},{\text{ for }}x>\lambda ,}$$ where ${\displaystyle \operatorname {D} _{\text{KL}}(Q\parallel P)}$ is the Kullback–Leibler divergence of ${\displaystyle Q=\operatorname {Pois} (x)}$ from ${\displaystyle P=\operatorname {Pois} (\lambda )}$.
• Inequalities that relate the distribution function of a Poisson random variable ${\displaystyle X\sim \operatorname {Pois} (\lambda )}$ to the Standard normal distribution function ${\displaystyle \Phi (x)}$ are as follows:^[34]

$${\displaystyle \Phi {\left(\operatorname {sign} (k-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}}\right)}<P(X\leq k)<\Phi {\left(\operatorname {sign} (k+1-\lambda ){\sqrt {2\operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}}\right)},{\text{ for }}k>0,}$$ where ${\displaystyle \operatorname {D} _{\text{KL}}(Q_{-}\parallel P)}$ is the Kullback–Leibler divergence of ${\displaystyle Q_{-}=\operatorname {Pois} (k)}$ from ${\displaystyle P=\operatorname {Pois} (\lambda )}$ and ${\displaystyle \operatorname {D} _{\text{KL}}(Q_{+}\parallel P)}$ is the Kullback–Leibler divergence of ${\displaystyle Q_{+}=\operatorname {Pois} (k+1)}$ from ${\displaystyle P}$.

Let ${\displaystyle X\sim \operatorname {Pois} (\lambda )}$ and ${\displaystyle Y\sim \operatorname {Pois} (\mu )}$ be independent random variables, with ${\displaystyle \lambda <\mu ,}$ then we have that $${\displaystyle {\frac {e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}{(\lambda +\mu )^{2}}}-{\frac {e^{-(\lambda +\mu )}}{2{\sqrt {\lambda \mu }}}}-{\frac {e^{-(\lambda +\mu )}}{4\lambda \mu }}\leq P(X-Y\geq 0)\leq e^{-({\sqrt {\mu }}-{\sqrt {\lambda }})^{2}}}$$

The upper bound is proved using a standard Chernoff bound.

The lower bound can be proved by noting that ${\displaystyle P(X-Y\geq 0\mid X+Y=i)}$ is the probability that ${\textstyle Z\geq {\frac {i}{2}},}$ where ${\textstyle Z\sim \operatorname {Bin} \left(i,{\frac {\lambda }{\lambda +\mu }}\right),}$ which is bounded below by ${\textstyle {\frac {1}{(i+1)^{2}}}e^{-iD\left(0.5\|{\frac {\lambda }{\lambda +\mu }}\right)},}$ where ${\displaystyle D}$ is relative entropy (See the entry on bounds on tails of binomial distributions for details). Further noting that ${\displaystyle X+Y\sim \operatorname {Pois} (\lambda +\mu ),}$ and computing a lower bound on the unconditional probability gives the result. More details can be found in the appendix of Kamath et al.^[35]

The Poisson distribution can be derived as a limiting case to the binomial distribution as the number of trials goes to infinity and the expected number of successes remains fixed — see law of rare events below. Therefore, it can be used as an approximation of the binomial distribution if n is sufficiently large and p is sufficiently small. The Poisson distribution is a good approximation of the binomial distribution if n is at least 20 and p is smaller than or equal to 0.05, and an excellent approximation if n ≥ 100 and np ≤ 10.^[36] Letting ${\displaystyle F_{\mathrm {B} }}$ and ${\displaystyle F_{\mathrm {P} }}$ be the respective cumulative density functions of the binomial and Poisson distributions, one has: $${\displaystyle F_{\mathrm {B} }(k;n,p)\ \approx \ F_{\mathrm {P} }(k;\lambda =np).}$$ One derivation of this uses probability-generating functions.^[37] Consider a Bernoulli trial (coin-flip) whose probability of one success (or expected number of successes) is ${\displaystyle \lambda \leq 1}$ within a given interval. Split the interval into n parts, and perform a trial in each subinterval with probability ${\displaystyle {\tfrac {\lambda }{n}}}$. The probability of k successes out of n trials over the entire interval is then given by the binomial distribution $${\displaystyle p_{k}^{(n)}={\binom {n}{k}}\left({\frac {\lambda }{n}}\right)^{\!k}\left(1{-}{\frac {\lambda }{n}}\right)^{\!n-k},}$$ whose generating function is: $${\displaystyle P^{(n)}(x)=\sum _{k=0}^{n}p_{k}^{(n)}x^{k}=\left(1-{\frac {\lambda }{n}}+{\frac {\lambda }{n}}x\right)^{n}.}$$ Taking the limit as n increases to infinity (with x fixed) and applying the product limit definition of the exponential function, this reduces to the generating function of the Poisson distribution: $${\displaystyle \lim _{n\to \infty }P^{(n)}(x)=\lim _{n\to \infty }\left(1{+}{\tfrac {\lambda (x-1)}{n}}\right)^{n}=e^{\lambda (x-1)}=\sum _{k=0}^{\infty }e^{-\lambda }{\frac {\lambda ^{k}}{k!}}x^{k}.}$$

• If ${\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}$ and ${\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}$ are independent, then the difference ${\displaystyle Y=X_{1}-X_{2}}$ follows a Skellam distribution.
• If ${\displaystyle X_{1}\sim \mathrm {Pois} (\lambda _{1})\,}$ and ${\displaystyle X_{2}\sim \mathrm {Pois} (\lambda _{2})\,}$ are independent, then the distribution of ${\displaystyle X_{1}}$ conditional on ${\displaystyle X_{1}+X_{2}}$ is a binomial distribution.
  Specifically, if ${\displaystyle X_{1}+X_{2}=k,}$ then ${\displaystyle X_{1}|X_{1}+X_{2}=k\sim \mathrm {Binom} (k,\lambda _{1}/(\lambda _{1}+\lambda _{2})).}$
  More generally, if X₁, X₂, ..., X_n are independent Poisson random variables with parameters λ₁, λ₂, ..., λ_n then
  given ${\displaystyle \sum _{j=1}^{n}X_{j}=k,}$ it follows that ${\displaystyle X_{i}{\Big |}\sum _{j=1}^{n}X_{j}=k\sim \mathrm {Binom} \left(k,{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right).}$ In fact, ${\displaystyle \{X_{i}\}\sim \mathrm {Multinom} \left(k,\left\{{\frac {\lambda _{i}}{\sum _{j=1}^{n}\lambda _{j}}}\right\}\right).}$
• If ${\displaystyle X\sim \mathrm {Pois} (\lambda )\,}$ and the distribution of ${\displaystyle Y}$ conditional on X = k is a binomial distribution, ${\displaystyle Y\mid (X=k)\sim \mathrm {Binom} (k,p),}$ then the distribution of Y follows a Poisson distribution ${\displaystyle Y\sim \mathrm {Pois} (\lambda \cdot p).}$ In fact, if, conditional on ${\displaystyle \{X=k\},}$ ${\displaystyle \{Y_{i}\}}$ follows a multinomial distribution, ${\displaystyle \{Y_{i}\}\mid (X=k)\sim \mathrm {Multinom} \left(k,p_{i}\right),}$ then each ${\displaystyle Y_{i}}$ follows an independent Poisson distribution ${\displaystyle Y_{i}\sim \mathrm {Pois} (\lambda \cdot p_{i}),\rho (Y_{i},Y_{j})=0.}$
• The Poisson distribution is a special case of the discrete compound Poisson distribution (or stuttering Poisson distribution) with only a parameter.^[38][39] The discrete compound Poisson distribution can be deduced from the limiting distribution of univariate multinomial distribution. It is also a special case of a compound Poisson distribution.
• For sufficiently large values of λ, (say λ > 1000), the normal distribution with mean λ and variance λ (standard deviation ${\displaystyle {\sqrt {\lambda }}}$) is an excellent approximation to the Poisson distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate continuity correction is performed, i.e., if P(X ≤ x), where x is a non-negative integer, is replaced by P(X ≤ x + 0.5). $${\displaystyle F_{\mathrm {Poisson} }(x;\lambda )\approx F_{\mathrm {normal} }(x;\mu =\lambda ,\sigma ^{2}=\lambda )}$$
• Variance-stabilizing transformation: If ${\displaystyle X\sim \mathrm {Pois} (\lambda ),}$ then^[8]: 168 $${\displaystyle Y=2{\sqrt {X}}\approx {\mathcal {N}}(2{\sqrt {\lambda }};1),}$$ and^[40]: 196 $${\displaystyle Y={\sqrt {X}}\approx {\mathcal {N}}({\sqrt {\lambda }};1/4).}$$ Under this transformation, the convergence to normality (as ${\displaystyle \lambda }$ increases) is far faster than the untransformed variable.^{[citation needed]} Other, slightly more complicated, variance stabilizing transformations are available,^[8]: 168 one of which is Anscombe transform.^[41] See Data transformation (statistics) for more general uses of transformations.
• If for every t > 0 the number of arrivals in the time interval [0, t] follows the Poisson distribution with mean λt, then the sequence of inter-arrival times are independent and identically distributed exponential random variables having mean 1/λ.^{[42]: 317–319}
• The cumulative distribution functions of the Poisson and chi-squared distributions are related in the following ways:^[8]: 167 $${\displaystyle F_{\text{Poisson}}(k;\lambda )=1-F_{\chi ^{2}}(2\lambda ;2(k+1))\quad \quad {\text{ integer }}k,}$$ and^[8]: 158 $${\displaystyle P(X=k)=F_{\chi ^{2}}(2\lambda ;2(k+1))-F_{\chi ^{2}}(2\lambda ;2k).}$$

Assume ${\displaystyle X_{1}\sim \operatorname {Pois} (\lambda _{1}),X_{2}\sim \operatorname {Pois} (\lambda _{2}),\dots ,X_{n}\sim \operatorname {Pois} (\lambda _{n})}$ where ${\displaystyle \lambda _{1}+\lambda _{2}+\dots +\lambda _{n}=1,}$ then^[43] ${\displaystyle (X_{1},X_{2},\dots ,X_{n})}$ is multinomially distributed ${\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Mult} (N,\lambda _{1},\lambda _{2},\dots ,\lambda _{n})}$ conditioned on ${\displaystyle N=X_{1}+X_{2}+\dots X_{n}.}$

This means^{[32]: 101-102}, among other things, that for any nonnegative function ${\displaystyle f(x_{1},x_{2},\dots ,x_{n}),}$ if ${\displaystyle (Y_{1},Y_{2},\dots ,Y_{n})\sim \operatorname {Mult} (m,\mathbf {p} )}$ is multinomially distributed, then $${\displaystyle \operatorname {E} [f(Y_{1},Y_{2},\dots ,Y_{n})]\leq e{\sqrt {m}}\operatorname {E} [f(X_{1},X_{2},\dots ,X_{n})]}$$ where ${\displaystyle (X_{1},X_{2},\dots ,X_{n})\sim \operatorname {Pois} (\mathbf {p} ).}$

The factor of ${\displaystyle e{\sqrt {m}}}$ can be replaced by 2 if ${\displaystyle f}$ is further assumed to be monotonically increasing or decreasing.

This distribution has been extended to the bivariate case.^[44] The generating function for this distribution is $${\displaystyle g(u,v)=\exp[(\theta _{1}-\theta _{12})(u-1)+(\theta _{2}-\theta _{12})(v-1)+\theta _{12}(uv-1)]}$$ with $${\displaystyle \theta _{1},\theta _{2}>\theta _{12}>0}$$

The marginal distributions are Poisson(θ₁) and Poisson(θ₂) and the correlation coefficient is limited to the range $${\displaystyle 0\leq \rho \leq \min \left\{{\sqrt {\frac {\theta _{1}}{\theta _{2}}}},{\sqrt {\frac {\theta _{2}}{\theta _{1}}}}\right\}}$$

A simple way to generate a bivariate Poisson distribution ${\displaystyle X_{1},X_{2}}$ is to take three independent Poisson distributions ${\displaystyle Y_{1},Y_{2},Y_{3}}$ with means ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}$ and then set ${\displaystyle X_{1}=Y_{1}+Y_{3},X_{2}=Y_{2}+Y_{3}.}$ The probability function of the bivariate Poisson distribution is $${\displaystyle \Pr(X_{1}=k_{1},X_{2}=k_{2})=\exp \left(-\lambda _{1}-\lambda _{2}-\lambda _{3}\right){\frac {\lambda _{1}^{k_{1}}}{k_{1}!}}{\frac {\lambda _{2}^{k_{2}}}{k_{2}!}}\sum _{k=0}^{\min(k_{1},k_{2})}{\binom {k_{1}}{k}}{\binom {k_{2}}{k}}k!\left({\frac {\lambda _{3}}{\lambda _{1}\lambda _{2}}}\right)^{k}}$$

The free Poisson distribution^[45] with jump size ${\displaystyle \alpha }$ and rate ${\displaystyle \lambda }$ arises in free probability theory as the limit of repeated free convolution $${\displaystyle \left(\left(1-{\frac {\lambda }{N}}\right)\delta _{0}+{\frac {\lambda }{N}}\delta _{\alpha }\right)^{\boxplus N}}$$ as N → ∞.

In other words, let ${\displaystyle X_{N}}$ be random variables so that ${\displaystyle X_{N}}$ has value ${\displaystyle \alpha }$ with probability ${\textstyle {\frac {\lambda }{N}}}$ and value 0 with the remaining probability. Assume also that the family ${\displaystyle X_{1},X_{2},\ldots }$ are freely independent. Then the limit as ${\displaystyle N\to \infty }$ of the law of ${\displaystyle X_{1}+\cdots +X_{N}}$ is given by the Free Poisson law with parameters ${\displaystyle \lambda ,\alpha .}$

This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.

The measure associated to the free Poisson law is given by^[46] $${\displaystyle \mu ={\begin{cases}(1-\lambda )\delta _{0}+\nu ,&{\text{if }}0\leq \lambda \leq 1\\\nu ,&{\text{if }}\lambda >1,\end{cases}}}$$ where $${\displaystyle \nu ={\frac {1}{2\pi \alpha t}}{\sqrt {4\lambda \alpha ^{2}-(t-\alpha (1+\lambda ))^{2}}}\,dt}$$ and has support ${\displaystyle [\alpha (1-{\sqrt {\lambda }})^{2},\alpha (1+{\sqrt {\lambda }})^{2}].}$

This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to ${\displaystyle \kappa _{n}=\lambda \alpha ^{n}.}$

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher^[47]

The R-transform of the free Poisson law is given by $${\displaystyle R(z)={\frac {\lambda \alpha }{1-\alpha z}}.}$$

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by $${\displaystyle G(z)={\frac {z+\alpha -\lambda \alpha -{\sqrt {(z-\alpha (1+\lambda ))^{2}-4\lambda \alpha ^{2}}}}{2\alpha z}}}$$

The S-transform is given by $${\displaystyle S(z)={\frac {1}{z+\lambda }}}$$ in the case that ${\displaystyle \alpha =1.}$

Given a sample of n measured values ${\displaystyle k_{i}\in \{0,1,\dots \},}$ for i = 1, ..., n, we wish to estimate the value of the parameter λ of the Poisson population from which the sample was drawn. The maximum likelihood estimate is^[48]

$${\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }={\frac {1}{n}}\sum _{i=1}^{n}k_{i}\ .}$$

Since each observation has expectation λ, so does the sample mean. Therefore, the maximum likelihood estimate is an unbiased estimator of λ. It is also an efficient estimator since its variance achieves the Cramér–Rao lower bound (CRLB).^[49] Hence it is minimum-variance unbiased. Also it can be proven that the sum (and hence the sample mean as it is a one-to-one function of the sum) is a complete and sufficient statistic for λ.

To prove sufficiency we may use the factorization theorem. Consider partitioning the probability mass function of the joint Poisson distribution for the sample into two parts: one that depends solely on the sample ${\displaystyle \mathbf {x} }$, called ${\displaystyle h(\mathbf {x} )}$, and one that depends on the parameter ${\displaystyle \lambda }$ and the sample ${\displaystyle \mathbf {x} }$ only through the function ${\displaystyle T(\mathbf {x} ).}$ Then ${\displaystyle T(\mathbf {x} )}$ is a sufficient statistic for ${\displaystyle \lambda .}$

$${\displaystyle P(\mathbf {x} )=\prod _{i=1}^{n}{\frac {\lambda ^{x_{i}}e^{-\lambda }}{x_{i}!}}={\frac {1}{\prod _{i=1}^{n}x_{i}!}}\times \lambda ^{\sum _{i=1}^{n}x_{i}}e^{-n\lambda }}$$

The first term ${\displaystyle h(\mathbf {x} )}$ depends only on ${\displaystyle \mathbf {x} }$. The second term ${\displaystyle g(T(\mathbf {x} )|\lambda )}$ depends on the sample only through ${\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}x_{i}.}$ Thus, ${\displaystyle T(\mathbf {x} )}$ is sufficient.

To find the parameter λ that maximizes the probability function for the Poisson population, we can use the logarithm of the likelihood function:

$${\displaystyle {\begin{aligned}\ell (\lambda )&=\ln \prod _{i=1}^{n}f(k_{i}\mid \lambda )\\&=\sum _{i=1}^{n}\ln \!\left({\frac {e^{-\lambda }\lambda ^{k_{i}}}{k_{i}!}}\right)\\&=-n\lambda +\left(\sum _{i=1}^{n}k_{i}\right)\ln(\lambda )-\sum _{i=1}^{n}\ln(k_{i}!).\end{aligned}}}$$

We take the derivative of ${\displaystyle \ell }$ with respect to λ and compare it to zero:

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \lambda }}\ell (\lambda )=0\iff -n+\left(\sum _{i=1}^{n}k_{i}\right){\frac {1}{\lambda }}=0.\!}$$

Solving for λ gives a stationary point.

$${\displaystyle \lambda ={\frac {\sum _{i=1}^{n}k_{i}}{n}}}$$

So λ is the average of the k_i values. Obtaining the sign of the second derivative of L at the stationary point will determine what kind of extreme value λ is.

$${\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-\lambda ^{-2}\sum _{i=1}^{n}k_{i}}$$

Evaluating the second derivative at the stationary point gives:

$${\displaystyle {\frac {\partial ^{2}\ell }{\partial \lambda ^{2}}}=-{\frac {n^{2}}{\sum _{i=1}^{n}k_{i}}}}$$

which is the negative of n times the reciprocal of the average of the k_i. This expression is negative when the average is positive. If this is satisfied, then the stationary point maximizes the probability function.

For completeness, a family of distributions is said to be complete if and only if ${\displaystyle E(g(T))=0}$ implies that ${\displaystyle P_{\lambda }(g(T)=0)=1}$ for all ${\displaystyle \lambda .}$ If the individual ${\displaystyle X_{i}}$ are iid ${\displaystyle \mathrm {Po} (\lambda ),}$ then ${\textstyle T(\mathbf {x} )=\sum _{i=1}^{n}X_{i}\sim \mathrm {Po} (n\lambda ).}$ Knowing the distribution we want to investigate, it is easy to see that the statistic is complete.

$${\displaystyle E(g(T))=\sum _{t=0}^{\infty }g(t){\frac {(n\lambda )^{t}e^{-n\lambda }}{t!}}=0}$$

For this equality to hold, ${\displaystyle g(t)}$ must be 0. This follows from the fact that none of the other terms will be 0 for all ${\displaystyle t}$ in the sum and for all possible values of ${\displaystyle \lambda .}$ Hence, ${\displaystyle E(g(T))=0}$ for all ${\displaystyle \lambda }$ implies that ${\displaystyle P_{\lambda }(g(T)=0)=1,}$ and the statistic has been shown to be complete.

The confidence interval for the mean of a Poisson distribution can be expressed using the relationship between the cumulative distribution functions of the Poisson and chi-squared distributions. The chi-squared distribution is itself closely related to the gamma distribution, and this leads to an alternative expression. Given an observation k from a Poisson distribution with mean μ, a confidence interval for μ with confidence level 1 – α is

$${\displaystyle {\tfrac {1}{2}}\chi ^{2}(\alpha /2;2k)\leq \mu \leq {\tfrac {1}{2}}\chi ^{2}(1-\alpha /2;2k+2),}$$

or equivalently,

$${\displaystyle F^{-1}(\alpha /2;k,1)\leq \mu \leq F^{-1}(1-\alpha /2;k+1,1),}$$

where ${\displaystyle \chi ^{2}(p;n)}$ is the quantile function (corresponding to a lower tail area p) of the chi-squared distribution with n degrees of freedom and ${\displaystyle F^{-1}(p;n,1)}$ is the quantile function of a gamma distribution with shape parameter n and scale parameter 1.^{[8]: 176-178 [50]} This interval is 'exact' in the sense that its coverage probability is never less than the nominal 1 – α.

When quantiles of the gamma distribution are not available, an accurate approximation to this exact interval has been proposed (based on the Wilson–Hilferty transformation):^[51] $${\displaystyle k\left(1-{\frac {1}{9k}}-{\frac {z_{\alpha /2}}{3{\sqrt {k}}}}\right)^{3}\leq \mu \leq (k+1)\left(1-{\frac {1}{9(k+1)}}+{\frac {z_{\alpha /2}}{3{\sqrt {k+1}}}}\right)^{3},}$$ where ${\displaystyle z_{\alpha /2}}$ denotes the standard normal deviate with upper tail area α / 2.

For application of these formulae in the same context as above (given a sample of n measured values k_i each drawn from a Poisson distribution with mean λ), one would set

$${\displaystyle k=\sum _{i=1}^{n}k_{i},}$$ calculate an interval for μ = nλ, and then derive the interval for λ.

In Bayesian inference, the conjugate prior for the rate parameter λ of the Poisson distribution is the gamma distribution.^[52] Let

$${\displaystyle \lambda \sim \mathrm {Gamma} (\alpha ,\beta )}$$

denote that λ is distributed according to the gamma density g parameterized in terms of a shape parameter α and an inverse scale parameter β:

$${\displaystyle g(\lambda \mid \alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\;\lambda ^{\alpha -1}\;e^{-\beta \,\lambda }\qquad {\text{ for }}\lambda >0\,\!.}$$

Then, given the same sample of n measured values k_i as before, and a prior of Gamma(α, β), the posterior distribution is

$${\displaystyle \lambda \sim \mathrm {Gamma} {\left(\alpha +\sum _{i=1}^{n}k_{i},\beta +n\right)}.}$$

Note that the posterior mean is linear and is given by $${\displaystyle E[\lambda \mid k_{1},\ldots ,k_{n}]={\frac {\alpha +\sum _{i=1}^{n}k_{i}}{\beta +n}}.}$$ It can be shown that gamma distribution is the only prior that induces linearity of the conditional mean. Moreover, a converse result exists which states that if the conditional mean is close to a linear function in the ${\displaystyle L_{2}}$ distance than the prior distribution of λ must be close to gamma distribution in Levy distance.^[53]

The posterior mean E[λ] approaches the maximum likelihood estimate ${\displaystyle {\widehat {\lambda }}_{\mathrm {MLE} }}$ in the limit as ${\displaystyle \alpha \to 0,\beta \to 0,}$ which follows immediately from the general expression of the mean of the gamma distribution.

The posterior predictive distribution for a single additional observation is a negative binomial distribution,^[54]: 53 sometimes called a gamma–Poisson distribution.

Suppose ${\displaystyle X_{1},X_{2},\dots ,X_{p}}$ is a set of independent random variables from a set of ${\displaystyle p}$ Poisson distributions, each with a parameter ${\displaystyle \lambda _{i},}$ ${\displaystyle i=1,\dots ,p,}$ and we would like to estimate these parameters. Then, Clevenson and Zidek show that under the normalized squared error loss ${\textstyle L(\lambda ,{\hat {\lambda }})=\sum _{i=1}^{p}\lambda _{i}^{-1}({\hat {\lambda }}_{i}-\lambda _{i})^{2},}$ when ${\displaystyle p>1,}$ then, similar as in Stein's example for the Normal means, the MLE estimator ${\displaystyle {\hat {\lambda }}_{i}=X_{i}}$ is inadmissible.^[55]

In this case, a family of minimax estimators is given for any ${\displaystyle 0<c\leq 2(p-1)}$ and ${\displaystyle b\geq (p-2+p^{-1})}$ as^[56] $${\displaystyle {\hat {\lambda }}_{i}=\left(1-{\frac {c}{b+\sum _{i=1}^{p}X_{i}}}\right)X_{i},\qquad i=1,\dots ,p.}$$

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Some applications of the Poisson distribution to count data (number of events):^[57]

• telecommunication: telephone calls arriving in a system,
• astronomy: photons arriving at a telescope,
• chemistry: the molar mass distribution of a living polymerization,^[58]
• biology: the number of mutations on a strand of DNA per unit length,
• management: customers arriving at a counter or call centre,^[59]
• finance and insurance: number of losses or claims occurring in a given period of time,
• seismology: asymptotic Poisson model of risk for large earthquakes,^[60]
• radioactivity: decays in a given time interval in a radioactive sample,^[61]
• optics: number of photons emitted in a single laser pulse (a major vulnerability of quantum key distribution protocols, known as photon number splitting).

More examples of counting events that may be modelled as Poisson processes include:

• soldiers killed by horse-kicks each year in each corps in the Prussian cavalry. This example was used in a book by Ladislaus Bortkiewicz (1868–1931),^{[12]: 23-25}
• yeast cells used when brewing Guinness beer. This example was used by William Sealy Gosset (1876–1937),^[62][63]
• phone calls arriving at a call centre within a minute. This example was described by A.K. Erlang (1878–1929),^[64]
• goals in sports involving two competing teams,^[65]
• deaths per year in a given age group,^[66]
• jumps in a stock price in a given time interval,
• times a web server is accessed per minute (under an assumption of homogeneity),
• mutations in a given stretch of DNA after a certain amount of radiation,
• cells infected at a given multiplicity of infection,
• bacteria in a certain amount of liquid,^[67]
• photons arriving on a pixel circuit at a given illumination over a given time period,^[68]
• landing of V-1 flying bombs on London during World War II, investigated by R. D. Clarke in 1946.^[69]