Factorial moment generating function

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the factorial moment generating function (FMGF) of the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> of a <a href="/facts/Real_number/R02gw5Pb">real-valued</a> <a href="/facts/Random_variable/TwTBXnLT">random variable</a> X is defined as

M
          
            X
          
        
        (
        t
        )
        =
        E
        ⁡
        
          
            [
          
        
        
          t
          
            X
          
        
        
          
            ]
          
        
      
    
    {\displaystyle M_{X}(t)=\operatorname {E} {\bigl [}t^{X}{\bigr ]}}

for all <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> t for which this <a href="/facts/Expected_value/1XV0JKL8">expected value</a> exists. This is the case at least for all t on the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a> 
 
 
 
 
 |
 
 t
 
 |
 
 =
 1
 
 
 {\displaystyle |t|=1}
 
, see <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a>. If X is a discrete random variable taking values only in the set {0,1, ...} of non-negative <a href="/facts/Integer/PUwwrawx">integers</a>, then 
 
 
 
 
 M
 
 X
 
 
 
 
 {\displaystyle M_{X}}
 
 is also called <a href="/facts/Probability-generating_function/gm7Kxx4g">probability-generating function</a> (PGF) of X and 
 
 
 
 
 M
 
 X
 
 
 (
 t
 )
 
 
 {\displaystyle M_{X}(t)}
 
 is well-defined at least for all t on the <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Unit_disk/phrUu7yC">unit disk</a> 
 
 
 
 
 |
 
 t
 
 |
 
 ≤
 1
 
 
 {\displaystyle |t|\leq 1}
 
.
The factorial moment generating function generates the <a href="/facts/Factorial_moment/RDIIdnd8">factorial moments</a> of the <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a>.
Provided 
 
 
 
 
 M
 
 X
 
 
 
 
 {\displaystyle M_{X}}
 
 exists in a <a href="/facts/Neighbourhood_(mathematics)/XHKgrpkp">neighbourhood</a> of t = 1, the nth factorial moment is given by

E
        ⁡
        [
        (
        X
        
          )
          
            n
          
        
        ]
        =
        
          M
          
            X
          
          
            (
            n
            )
          
        
        (
        1
        )
        =
        
          
            
            
              
                
                  
                    d
                  
                  
                    n
                  
                
                
                  
                    d
                  
                  
                    t
                    
                      n
                    
                  
                
              
            
            |
          
          
            t
            =
            1
          
        
        
          M
          
            X
          
        
        (
        t
        )
        ,
      
    
    {\displaystyle \operatorname {E} [(X)_{n}]=M_{X}^{(n)}(1)=\left.{\frac {\mathrm {d} ^{n}}{\mathrm {d} t^{n}}}\right|_{t=1}M_{X}(t),}

where the <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbol</a> (x)n is the <a href="/facts/Falling_factorial/sAxnmGoC">falling factorial</a>

(
        x
        
          )
          
            n
          
        
        =
        x
        (
        x
        −
        1
        )
        (
        x
        −
        2
        )
        ⋯
        (
        x
        −
        n
        +
        1
        )
        .
        
      
    
    {\displaystyle (x)_{n}=x(x-1)(x-2)\cdots (x-n+1).\,}

(Many mathematicians, especially in the field of <a href="/facts/Special_function/JETSeucA">special functions</a>, use the same notation to represent the <a href="/facts/Rising_factorial/sAxnmGoC">rising factorial</a>.)

Factorial moment generating function open-in-new

Factorial moment generating function