Noncentral beta distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the noncentral beta distribution is a <a href="/facts/Continuous_probability_distribution/EpsKKVRu">continuous probability distribution</a> that is a <a href="/facts/Noncentral_distribution/uWgoRYTm">noncentral generalization</a> of the (central) <a href="/facts/Beta_distribution/DbJk4eeV">beta distribution</a>.
The noncentral beta distribution (Type I) is the distribution of the ratio

X
        =
        
          
            
              
                χ
                
                  m
                
                
                  2
                
              
              (
              λ
              )
            
            
              
                χ
                
                  m
                
                
                  2
                
              
              (
              λ
              )
              +
              
                χ
                
                  n
                
                
                  2
                
              
            
          
        
        ,
      
    
    {\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}

where 
 
 
 
 
 χ
 
 m
 
 
 2
 
 
 (
 λ
 )
 
 
 {\displaystyle \chi _{m}^{2}(\lambda )}
 
 is a <a href="/facts/Noncentral_chi-squared_distribution/saAcjyRh">
noncentral chi-squared</a> random variable with degrees of freedom m and noncentrality parameter 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
, and 
 
 
 
 
 χ
 
 n
 
 
 2
 
 
 
 
 {\displaystyle \chi _{n}^{2}}
 
 is a central <a href="/facts/Chi-squared_distribution/wYnWWgUe">chi-squared</a> random variable with degrees of freedom n, independent of 
 
 
 
 
 χ
 
 m
 
 
 2
 
 
 (
 λ
 )
 
 
 {\displaystyle \chi _{m}^{2}(\lambda )}
 
.
In this case, 
 
 
 
 X
 ∼
 
 
 Beta
 
 
 
 (
 
 
 
 m
 2
 
 
 ,
 
 
 n
 2
 
 
 ,
 λ
 
 )
 
 
 
 {\displaystyle X\sim {\mbox{Beta}}\left({\frac {m}{2}},{\frac {n}{2}},\lambda \right)}

A Type II noncentral beta distribution is the distribution
of the ratio

Y
        =
        
          
            
              χ
              
                n
              
              
                2
              
            
            
              
                χ
                
                  n
                
                
                  2
                
              
              +
              
                χ
                
                  m
                
                
                  2
                
              
              (
              λ
              )
            
          
        
        ,
      
    
    {\displaystyle Y={\frac {\chi _{n}^{2}}{\chi _{n}^{2}+\chi _{m}^{2}(\lambda )}},}

where the noncentral chi-squared variable is in the denominator only. If 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 follows 
the type II distribution, then 
 
 
 
 X
 =
 1
 −
 Y
 
 
 {\displaystyle X=1-Y}
 
 follows a type I distribution.

Noncentral beta distribution open-in-new

Noncentral beta distribution