Marsaglia polar method

The <a href="/facts/George_Marsaglia/Jlu1Defc">Marsaglia</a> polar method is a <a href="/facts/Pseudo-random_number_sampling/xIkyQJqu">pseudo-random number sampling</a> method for generating a pair of independent <a href="/facts/Standard_normal_random_variable/UapjjPyQ">standard normal random variables</a>.
Standard normal random variables are frequently used in <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, <a href="/facts/Computational_statistics/noW2SBX8">computational statistics</a>, and in particular, in applications of the <a href="/facts/Monte_Carlo_method/AkHjY7jc">Monte Carlo method</a>.
The polar method works by choosing random points (x, y) in the square −1 < x < 1, −1 < y < 1 until

0
 <
 s
 =
 
 x
 
 2
 
 
 +
 
 y
 
 2
 
 
 <
 1
 ,
 
 
 
 {\displaystyle 0<s=x^{2}+y^{2}<1,\,}

and then returning the required pair of normal <a href="/facts/Random_variable/TwTBXnLT">random variables</a> as

x
        
          
            
              
                −
                2
                ln
                ⁡
                (
                s
                )
              
              s
            
          
        
        
        ,
         
         
        y
        
          
            
              
                −
                2
                ln
                ⁡
                (
                s
                )
              
              s
            
          
        
        ,
      
    
    {\displaystyle x{\sqrt {\frac {-2\ln(s)}{s}}}\,,\ \ y{\sqrt {\frac {-2\ln(s)}{s}}},}

or, equivalently,

x
            
              s
            
          
        
        
          
            −
            2
            ln
            ⁡
            (
            s
            )
          
        
        
        ,
         
         
        
          
            y
            
              s
            
          
        
        
          
            −
            2
            ln
            ⁡
            (
            s
            )
          
        
        ,
      
    
    {\displaystyle {\frac {x}{\sqrt {s}}}{\sqrt {-2\ln(s)}}\,,\ \ {\frac {y}{\sqrt {s}}}{\sqrt {-2\ln(s)}},}

where 
 
 
 
 x
 
 /
 
 
 
 s
 
 
 
 
 {\displaystyle x/{\sqrt {s}}}
 
 and 
 
 
 
 y
 
 /
 
 
 
 s
 
 
 
 
 {\displaystyle y/{\sqrt {s}}}
 
 represent the <a href="/facts/Cosine/czej7ur0">cosine</a> and <a href="/facts/Sine/gQ6LXK8z">sine</a> of the angle that the vector (x, y) makes with x axis.

Marsaglia polar method open-in-new

Marsaglia polar method