---
title: "Displaced Poisson distribution"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Displaced_Poisson_distribution"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:22:26.926603+00:00"
instance: "kb-cron"
---

In statistics, the displaced Poisson, also known as the hyper-Poisson distribution, is a generalization of the Poisson distribution.


== Definitions ==


=== Probability mass function ===
The probability mass function is

  
    
      
        P
        (
        X
        =
        n
        )
        =
        
          
            {
            
              
                
                  
                    e
                    
                      −
                      λ
                    
                  
                  
                    
                      
                        
                          λ
                          
                            n
                            +
                            r
                          
                        
                        
                          
                            (
                            
                              n
                              +
                              r
                            
                            )
                          
                          !
                        
                      
                    
                  
                  ⋅
                  
                    
                      
                        1
                        
                          I
                          
                            (
                            
                              r
                              ,
                              λ
                            
                            )
                          
                        
                      
                    
                  
                  ,
                  
                  n
                  =
                  0
                  ,
                  1
                  ,
                  2
                  ,
                  …
                
                
                  
                    if 
                  
                  r
                  ≥
                  0
                
              
              
                
                  
                    e
                    
                      −
                      λ
                    
                  
                  
                    
                      
                        
                          λ
                          
                            n
                            +
                            r
                          
                        
                        
                          
                            (
                            
                              n
                              +
                              r
                            
                            )
                          
                          !
                        
                      
                    
                  
                  ⋅
                  
                    
                      
                        1
                        
                          I
                          
                            (
                            
                              r
                              +
                              s
                              ,
                              λ
                            
                            )
                          
                        
                      
                    
                  
                  ,
                  
                  n
                  =
                  s
                  ,
                  s
                  +
                  1
                  ,
                  s
                  +
                  2
                  ,
                  …
                
                
                  
                    otherwise
                  
                
              
            
            
          
        
      
    
    {\displaystyle P(X=n)={\begin{cases}e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r,\lambda \right)}},\quad n=0,1,2,\ldots &{\text{if }}r\geq 0\\[10pt]e^{-\lambda }{\dfrac {\lambda ^{n+r}}{\left(n+r\right)!}}\cdot {\dfrac {1}{I\left(r+s,\lambda \right)}},\quad n=s,s+1,s+2,\ldots &{\text{otherwise}}\end{cases}}}
  

where 
  
    
      
        λ
        >
        0
      
    
    {\displaystyle \lambda >0}
  
 and r is a new parameter; the Poisson distribution is recovered at r = 0.  Here 
  
    
      
        I
        
          (
          
            r
            ,
            λ
          
          )
        
      
    
    {\displaystyle I\left(r,\lambda \right)}
  
 is the Pearson's incomplete gamma function:

  
    
      
        I
        (
        r
        ,
        λ
        )
        =
        
          ∑
          
            y
            =
            r
          
          
            ∞
          
        
        
          
            
              
                e
                
                  −
                  λ
                
              
              
                λ
                
                  y
                
              
            
            
              y
              !
            
          
        
        ,
      
    
    {\displaystyle I(r,\lambda )=\sum _{y=r}^{\infty }{\frac {e^{-\lambda }\lambda ^{y}}{y!}},}
  

where s is the integral part of r. 
The motivation given by Staff is that the ratio of successive probabilities in the Poisson distribution (that is 
  
    
      
        P
        (
        X
        =
        n
        )
        
          /
        
        P
        (
        X
        =
        n
        −
        1
        )
      
    
    {\displaystyle P(X=n)/P(X=n-1)}
  
) is given by 
  
    
      
        λ
        
          /
        
        n
      
    
    {\displaystyle \lambda /n}
  
 for 
  
    
      
        n
        >
        0
      
    
    {\displaystyle n>0}
  
 and the displaced Poisson generalizes this ratio to 
  
    
      
        λ
        
          /
        
        
          (
          
            n
            +
            r
          
          )
        
      
    
    {\displaystyle \lambda /\left(n+r\right)}
  
.


=== Examples ===
One of the limitations of the Poisson distribution is that it assumes equidispersion – the mean and variance of the variable are equal. The displaced Poisson distribution may be useful to model underdispersed or overdispersed data, such as:

the distribution of insect populations in crop fields;
the number of flowers on plants;
motor vehicle crash counts; and
word or sentence lengths in writing.


== Properties ==


=== Descriptive Statistics ===
For a displaced Poisson-distributed random variable, the mean is equal to 
  
    
      
        λ
        −
        r
      
    
    {\displaystyle \lambda -r}
  
 and the variance is equal to 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
.
The mode of a displaced Poisson-distributed random variable are the integer values bounded by 
  
    
      
        λ
        −
        r
        −
        1
      
    
    {\displaystyle \lambda -r-1}
  
 and 
  
    
      
        λ
        −
        r
      
    
    {\displaystyle \lambda -r}
  
 when 
  
    
      
        λ
        ≥
        r
        +
        1
      
    
    {\displaystyle \lambda \geq r+1}
  
. When 
  
    
      
        λ
        <
        r
        +
        1
      
    
    {\displaystyle \lambda <r+1}
  
, there is a single mode at 
  
    
      
        x
        =
        0
      
    
    {\displaystyle x=0}
  
.
The first cumulant 
  
    
      
        
          κ
          
            1
          
        
      
    
    {\displaystyle \kappa _{1}}
  
 is equal to 
  
    
      
        λ
        −
        r
      
    
    {\displaystyle \lambda -r}
  
 and all subsequent cumulants 
  
    
      
        
          κ
          
            n
          
        
        ,
        n
        ≥
        2
      
    
    {\displaystyle \kappa _{n},n\geq 2}
  
 are equal to 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
.


== References ==