Inverse Gaussian distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of <a href="/facts/Continuous_probability_distribution/EpsKKVRu">continuous probability distributions</a> with <a href="/facts/Support_(mathematics)/BTuMGbsb">support</a> on (0,∞).
Its <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> is given by

f
        (
        x
        ;
        μ
        ,
        λ
        )
        =
        
          
            
              λ
              
                2
                π
                
                  x
                  
                    3
                  
                
              
            
          
        
        exp
        ⁡
        
          
            (
          
        
        −
        
          
            
              λ
              (
              x
              −
              μ
              
                )
                
                  2
                
              
            
            
              2
              
                μ
                
                  2
                
              
              x
            
          
        
        
          
            )
          
        
      
    
    {\displaystyle f(x;\mu ,\lambda )={\sqrt {\frac {\lambda }{2\pi x^{3}}}}\exp {\biggl (}-{\frac {\lambda (x-\mu )^{2}}{2\mu ^{2}x}}{\biggr )}}

for x > 0, where 
 
 
 
 μ
 >
 0
 
 
 {\displaystyle \mu >0}
 
 is the mean and 
 
 
 
 λ
 >
 0
 
 
 {\displaystyle \lambda >0}
 
 is the shape parameter.
The inverse Gaussian distribution has several properties analogous to a Gaussian distribution. The name can be misleading: it is an inverse only in that, while the Gaussian describes a <a href="/facts/Wiener_process/KPh7vYd7">Brownian motion's</a> level at a fixed time, the inverse Gaussian describes the distribution of the time a Brownian motion with positive drift takes to reach a fixed positive level.
Its <a href="/facts/Cumulant_generating_function/crLC9QLE">cumulant generating function</a> (logarithm of the characteristic function)[<a href="/facts/Cumulant/crLC9QLE">contradictory</a>] is the inverse of the cumulant generating function of a Gaussian random variable.
To indicate that a <a href="/facts/Random_variable/TwTBXnLT">random variable</a> X is inverse Gaussian-distributed with mean μ and shape parameter λ we write 
 
 
 
 X
 ∼
 IG
 ⁡
 (
 μ
 ,
 λ
 )
 
 
 
 
 {\displaystyle X\sim \operatorname {IG} (\mu ,\lambda )\,\!}
 
.

Inverse Gaussian distribution open-in-new

Inverse Gaussian distribution