Mixed Poisson distribution

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Notation	Pois ⁡ ( λ ) ∧ λ π ( λ ) {\displaystyle \operatorname {Pois} (\lambda )\,{\underset {\lambda }{\wedge }}\,\pi (\lambda )} $\operatorname {Pois} (\lambda )\,{\underset {\lambda }{\wedge }}\,\pi (\lambda )$
Parameters	λ ∈ ( 0 , ∞ ) {\displaystyle \lambda \in (0,\infty )} $\lambda \in (0,\infty )$
Support	k ∈ N 0 {\displaystyle k\in \mathbb {N} _{0}} $k\in \mathbb {N} _{0}$
PMF	∫ 0 ∞ λ k k ! e − λ π ( λ ) d λ {\displaystyle \int \limits _{0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda } $\int \limits _{0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda$
Mean	∫ 0 ∞ λ π ( λ ) d λ {\displaystyle \int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )\,d\lambda } $\int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )\,d\lambda$
Variance	∫ 0 ∞ ( λ + ( λ − μ π ) 2 ) π ( λ ) d λ {\displaystyle \int \limits _{0}^{\infty }(\lambda +(\lambda -\mu _{\pi })^{2})\,\,\pi (\lambda )\,d\lambda } $\int \limits _{0}^{\infty }(\lambda +(\lambda -\mu _{\pi })^{2})\,\,\pi (\lambda )\,d\lambda$
Skewness	( μ π + σ π 2 ) − 3 / 2 π ( λ ) d λ + μ π ] {\displaystyle {\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-3/2}\,{\Biggl \,\pi (\lambda )\,d{\lambda }+\mu _{\pi }{\Biggr ]}} ${\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-3/2}\,{\Biggl \,\pi (\lambda )\,d{\lambda }+\mu _{\pi }{\Biggr ]}$
MGF	M π ( e t − 1 ) {\displaystyle M_{\pi }(e^{t}-1)} $M_{\pi }(e^{t}-1)$ , with M π {\displaystyle M_{\pi }} $M_{\pi }$ the MGF of π
CF	M π ( e i t − 1 ) {\displaystyle M_{\pi }(e^{it}-1)} $M_{\pi }(e^{it}-1)$
PGF	M π ( z − 1 ) {\displaystyle M_{\pi }(z-1)} $M_{\pi }(z-1)$

A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution, and that the rate parameter itself is considered as a random variable. Hence it is a special case of a compound probability distribution. Mixed Poisson distributions can be found in actuarial mathematics as a general approach for the distribution of the number of claims and is also examined as an epidemiological model. It should not be confused with compound Poisson distribution or compound Poisson process.

Definition

A random variable X satisfies the mixed Poisson distribution with density π(λ) if it has the probability distribution

P ⁡ ( X = k ) = ∫ 0 ∞ λ k k ! e − λ π ( λ ) d λ . {\displaystyle \operatorname {P} (X=k)=\int _{0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda .}

\operatorname {P} (X=k)=\int _{0}^{\infty }{\frac {\lambda ^{k}}{k!}}e^{-\lambda }\,\,\pi (\lambda )\,\mathrm {d} \lambda .

If we denote the probabilities of the Poisson distribution by qλ(k), then

P ⁡ ( X = k ) = ∫ 0 ∞ q λ ( k ) π ( λ ) d λ . {\displaystyle \operatorname {P} (X=k)=\int _{0}^{\infty }q_{\lambda }(k)\,\,\pi (\lambda )\,\mathrm {d} \lambda .}

\operatorname {P} (X=k)=\int _{0}^{\infty }q_{\lambda }(k)\,\,\pi (\lambda )\,\mathrm {d} \lambda .

Properties

The variance is always bigger than the expected value. This property is called overdispersion. This is in contrast to the Poisson distribution where mean and variance are the same.
In practice, almost only densities of gamma distributions, logarithmic normal distributions and inverse Gaussian distributions are used as densities π(λ). If we choose the density of the gamma distribution, we get the negative binomial distribution, which explains why this is also called the Poisson gamma distribution.

In the following let μ π = ∫ 0 ∞ λ π ( λ ) d λ {\displaystyle \mu _{\pi }=\int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )\,d\lambda \,} $\mu _{\pi }=\int \limits _{0}^{\infty }\lambda \,\,\pi (\lambda )\,d\lambda \,$ be the expected value of the density π ( λ ) {\displaystyle \pi (\lambda )\,} $\pi (\lambda )\,$ and σ π 2 = ∫ 0 ∞ ( λ − μ π ) 2 π ( λ ) d λ {\displaystyle \sigma _{\pi }^{2}=\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{2}\,\,\pi (\lambda )\,d\lambda \,} $\sigma _{\pi }^{2}=\int \limits _{0}^{\infty }(\lambda -\mu _{\pi })^{2}\,\,\pi (\lambda )\,d\lambda \,$ be the variance of the density.

Expected value

The expected value of the mixed Poisson distribution is

E ⁡ ( X ) = μ π . {\displaystyle \operatorname {E} (X)=\mu _{\pi }.}

\operatorname {E} (X)=\mu _{\pi }.

Variance

For the variance one gets

Var ⁡ ( X ) = μ π + σ π 2 . {\displaystyle \operatorname {Var} (X)=\mu _{\pi }+\sigma _{\pi }^{2}.}

\operatorname {Var} (X)=\mu _{\pi }+\sigma _{\pi }^{2}.

Skewness

The skewness can be represented as

v ⁡ ( X ) = ( μ π + σ π 2 ) − 3 / 2 . {\displaystyle \operatorname {v} (X)={\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-3/2}\,{\Biggl }.}

\operatorname {v} (X)={\Bigl (}\mu _{\pi }+\sigma _{\pi }^{2}{\Bigr )}^{-3/2}\,{\Biggl }.

Characteristic function

The characteristic function has the form

φ X ( s ) = M π ( e i s − 1 ) . {\displaystyle \varphi _{X}(s)=M_{\pi }(e^{is}-1).\,}

\varphi _{X}(s)=M_{\pi }(e^{is}-1).\,

Where M π {\displaystyle M_{\pi }} $M_{\pi }$ is the moment generating function of the density.

Probability generating function

For the probability generating function, one obtains

m X ( s ) = M π ( s − 1 ) . {\displaystyle m_{X}(s)=M_{\pi }(s-1).\,}

m_{X}(s)=M_{\pi }(s-1).\,

Moment-generating function

The moment-generating function of the mixed Poisson distribution is

M X ( s ) = M π ( e s − 1 ) . {\displaystyle M_{X}(s)=M_{\pi }(e^{s}-1).\,}

M_{X}(s)=M_{\pi }(e^{s}-1).\,

Examples

Theorem — Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.

Proof

Let π ( λ ) = ( p 1 − p ) r Γ ( r ) λ r − 1 e − p 1 − p λ {\displaystyle \pi (\lambda )={\frac {({\frac {p}{1-p}})^{r}}{\Gamma (r)}}\lambda ^{r-1}e^{-{\frac {p}{1-p}}\lambda }} $\pi (\lambda )={\frac {({\frac {p}{1-p}})^{r}}{\Gamma (r)}}\lambda ^{r-1}e^{-{\frac {p}{1-p}}\lambda }$ be a density of a Γ ⁡ ( r , p 1 − p ) {\displaystyle \operatorname {\Gamma } \left(r,{\frac {p}{1-p}}\right)} $\operatorname {\Gamma } \left(r,{\frac {p}{1-p}}\right)$ distributed random variable.

P ⁡ ( X = k ) = 1 k ! ∫ 0 ∞ λ k e − λ ( p 1 − p ) r Γ ( r ) λ r − 1 e − p 1 − p λ d λ = p r ( 1 − p ) − r Γ ( r ) k ! ∫ 0 ∞ λ k + r − 1 e − λ 1 1 − p d λ = p r ( 1 − p ) − r Γ ( r ) k ! ( 1 − p ) k + r ∫ 0 ∞ λ k + r − 1 e − λ d λ ⏟ = Γ ( r + k ) = Γ ( r + k ) Γ ( r ) k ! ( 1 − p ) k p r {\displaystyle {\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!}}\int _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {({\frac {p}{1-p}})^{r}}{\Gamma (r)}}\lambda ^{r-1}e^{-{\frac {p}{1-p}}\lambda }\,\mathrm {d} \lambda \\&={\frac {p^{r}(1-p)^{-r}}{\Gamma (r)k!}}\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda {\frac {1}{1-p}}}\,\mathrm {d} \lambda \\&={\frac {p^{r}(1-p)^{-r}}{\Gamma (r)k!}}(1-p)^{k+r}\underbrace {\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda }\,\mathrm {d} \lambda } _{=\Gamma (r+k)}\\&={\frac {\Gamma (r+k)}{\Gamma (r)k!}}(1-p)^{k}p^{r}\end{aligned}}} ${\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!}}\int _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {({\frac {p}{1-p}})^{r}}{\Gamma (r)}}\lambda ^{r-1}e^{-{\frac {p}{1-p}}\lambda }\,\mathrm {d} \lambda \\&={\frac {p^{r}(1-p)^{-r}}{\Gamma (r)k!}}\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda {\frac {1}{1-p}}}\,\mathrm {d} \lambda \\&={\frac {p^{r}(1-p)^{-r}}{\Gamma (r)k!}}(1-p)^{k+r}\underbrace {\int _{0}^{\infty }\lambda ^{k+r-1}e^{-\lambda }\,\mathrm {d} \lambda } _{=\Gamma (r+k)}\\&={\frac {\Gamma (r+k)}{\Gamma (r)k!}}(1-p)^{k}p^{r}\end{aligned}}$

Therefore we get X ∼ NegB ⁡ ( r , p ) . {\displaystyle X\sim \operatorname {NegB} (r,p).} $X\sim \operatorname {NegB} (r,p).$

Theorem — Compounding a Poisson distribution with rate parameter distributed according to a exponential distribution yields a geometric distribution.

Proof

Let π ( λ ) = 1 β e − λ β {\displaystyle \pi (\lambda )={\frac {1}{\beta }}e^{-{\frac {\lambda }{\beta }}}} $\pi (\lambda )={\frac {1}{\beta }}e^{-{\frac {\lambda }{\beta }}}$ be a density of a E x p ( 1 β ) {\displaystyle \mathrm {Exp} \left({\frac {1}{\beta }}\right)} $\mathrm {Exp} \left({\frac {1}{\beta }}\right)$ distributed random variable. Using integration by parts n times yields:

P ⁡ ( X = k ) = 1 k ! ∫ 0 ∞ λ k e − λ 1 β e − λ β d λ = 1 k ! β ∫ 0 ∞ λ k e − λ ( 1 + β β ) d λ = 1 k ! β ⋅ k ! ( β 1 + β ) k ∫ 0 ∞ e − λ ( 1 + β β ) d λ = ( β 1 + β ) k ( 1 1 + β ) {\displaystyle {\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!}}\int \limits _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {1}{\beta }}e^{-{\frac {\lambda }{\beta }}}\,\mathrm {d} \lambda \\&={\frac {1}{k!\beta }}\int \limits _{0}^{\infty }\lambda ^{k}e^{-\lambda \left({\frac {1+\beta }{\beta }}\right)}\,\mathrm {d} \lambda \\&={\frac {1}{k!\beta }}\cdot k!\left({\frac {\beta }{1+\beta }}\right)^{k}\int \limits _{0}^{\infty }e^{-\lambda \left({\frac {1+\beta }{\beta }}\right)}\,\mathrm {d} \lambda \\&=\left({\frac {\beta }{1+\beta }}\right)^{k}\left({\frac {1}{1+\beta }}\right)\end{aligned}}}

{\begin{aligned}\operatorname {P} (X=k)&={\frac {1}{k!}}\int \limits _{0}^{\infty }\lambda ^{k}e^{-\lambda }{\frac {1}{\beta }}e^{-{\frac {\lambda }{\beta }}}\,\mathrm {d} \lambda \\&={\frac {1}{k!\beta }}\int \limits _{0}^{\infty }\lambda ^{k}e^{-\lambda \left({\frac {1+\beta }{\beta }}\right)}\,\mathrm {d} \lambda \\&={\frac {1}{k!\beta }}\cdot k!\left({\frac {\beta }{1+\beta }}\right)^{k}\int \limits _{0}^{\infty }e^{-\lambda \left({\frac {1+\beta }{\beta }}\right)}\,\mathrm {d} \lambda \\&=\left({\frac {\beta }{1+\beta }}\right)^{k}\left({\frac {1}{1+\beta }}\right)\end{aligned}}

Therefore we get X ∼ G e o ( 1 1 + β ) . {\displaystyle X\sim \operatorname {Geo\left({\frac {1}{1+\beta }}\right)} .}

X\sim \operatorname {Geo\left({\frac {1}{1+\beta }}\right)} .

Table of mixed Poisson distributions

mixing distribution	mixed Poisson distribution
gamma	negative binomial
exponential	geometric
inverse Gaussian	Sichel
Poisson	Neyman
generalized inverse Gaussian	Poisson-generalized inverse Gaussian
generalized gamma	Poisson-generalized gamma
generalized Pareto	Poisson-generalized Pareto
inverse-gamma	Poisson-inverse gamma
log-normal	Poisson-log-normal
Lomax	Poisson–Lomax
Pareto	Poisson–Pareto
Pearson’s family of distributions	Poisson–Pearson family
truncated normal	Poisson-truncated normal
uniform	Poisson-uniform
shifted gamma	Delaporte
beta with specific parameter values	Yule

Literature

Jan Grandell: Mixed Poisson Processes. Chapman & Hall, London 1997, ISBN 0-412-78700-8 .
Tom Britton: Stochastic Epidemic Models with Inference. Springer, 2019, doi:10.1007/978-3-030-30900-8

References

^ Willmot, Gordon E.; Lin, X. Sheldon (2001), "Mixed Poisson distributions", Lundberg Approximations for Compound Distributions with Insurance Applications, Lecture Notes in Statistics, vol. 156, New York, NY: Springer New York, pp. 37–49, doi:10.1007/978-1-4613-0111-0_3, ISBN 978-0-387-95135-5, retrieved 2022-07-08
^ Willmot, Gord (1986). "Mixed Compound Poisson Distributions". ASTIN Bulletin. 16 (S1): S59–S79. doi:10.1017/S051503610001165X. ISSN 0515-0361.
^ a b c d Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Astin Bulletin. 16: 5–7. doi:10.1017/S051503610001165X. S2CID 17737506.
^ Karlis, Dimitris; Xekalaki, Evdokia (2005). "Mixed Poisson Distributions". International Statistical Review. 73 (1): 35–58. doi:10.1111/j.1751-5823.2005.tb00250.x. ISSN 0306-7734. JSTOR 25472639. S2CID 53637483.

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's SU Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
multinomial
- Dirichlet
- negative
Continuous:
Dirichlet
- generalized
multivariate Laplace
multivariate normal
multivariate stable
multivariate t
normal-gamma
- inverse
Matrix-valued:
LKJ
matrix normal
matrix t
matrix gamma
- inverse
Wishart
- normal
- inverse
- normal-inverse
- complex

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate
and singular

Degenerate Dirac delta function Singular Cantor

Families