Geometric Poisson distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the geometric Poisson distribution (also called the Pólya–Aeppli distribution) is used for describing objects that come in clusters, where the number of clusters follows a <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> and the number of objects within a cluster follows a <a href="/facts/Geometric_distribution/c8wjfaoV">geometric distribution</a>. It is a particular case of the <a href="/facts/Compound_Poisson_distribution/tLHJbhIP">compound Poisson distribution</a>. 
The <a href="/facts/Probability_mass_function/LhurokRt">probability mass function</a> of a random variable N distributed according to the geometric Poisson distribution 
 
 
 
 
 
 P
 G
 
 
 (
 λ
 ,
 θ
 )
 
 
 {\displaystyle {\mathcal {PG}}(\lambda ,\theta )}
 
 is given by

f
          
            N
          
        
        (
        n
        )
        =
        
          P
          r
        
        (
        N
        =
        n
        )
        =
        
          
            {
            
              
                
                  
                    ∑
                    
                      k
                      =
                      1
                    
                    
                      n
                    
                  
                  
                    e
                    
                      −
                      λ
                    
                  
                  
                    
                      
                        λ
                        
                          k
                        
                      
                      
                        k
                        !
                      
                    
                  
                  (
                  1
                  −
                  θ
                  
                    )
                    
                      n
                      −
                      k
                    
                  
                  
                    θ
                    
                      k
                    
                  
                  
                    
                      
                        (
                      
                      
                        
                          n
                          −
                          1
                        
                        
                          k
                          −
                          1
                        
                      
                      
                        )
                      
                    
                  
                  ,
                
                
                  n
                  >
                  0
                
              
              
                
                  
                    e
                    
                      −
                      λ
                    
                  
                  ,
                
                
                  n
                  =
                  0
                
              
            
            
          
        
      
    
    {\displaystyle f_{N}(n)=\mathrm {Pr} (N=n)={\begin{cases}\sum _{k=1}^{n}e^{-\lambda }{\frac {\lambda ^{k}}{k!}}(1-\theta )^{n-k}\theta ^{k}{\binom {n-1}{k-1}},&n>0\\e^{-\lambda },&n=0\end{cases}}}

where λ is the parameter of the underlying <a href="/facts/Poisson_distribution/CvCzXkHr">Poisson distribution</a> and θ is the parameter of the geometric distribution.
The distribution was described by <a href="/facts/George_P%25C3%25B3lya/cYgGdvJu">George Pólya</a> in 1930. Pólya credited his student <a href="/facts/Alfred_Aeppli/2j7NXTgq">Alfred Aeppli</a>'s 1924 dissertation as the original source. It was called the geometric Poisson distribution by Sherbrooke in 1968, who gave probability tables with a precision of four decimal places.
The geometric Poisson distribution has been used to describe systems modelled by a <a href="/facts/Markov_model/Mlndun4Q">Markov model</a>, such as biological processes or traffic accidents.

Geometric Poisson distribution open-in-new

Geometric Poisson distribution