page_title: Łukaszyk–Karmowski metric
section_title: Continuous random variables
hierarchical_section_title: Łukaszyk–Karmowski metric / Continuous random variables
caption_attribution_description: Polski: μx − μy
context_page_description: In mathematics, the Łukaszyk–Karmowski metric is a function defining a distance between two random variables or two random vectors. This function is not a metric as it does not satisfy the identity of indiscernibles condition of the metric, that is for two identical arguments its value is greater than zero. The concept is named after Szymon Łukaszyk and Wojciech Karmowski.
context_section_description: The Łukaszyk–Karmowski metric D between two continuous independent random variables X and Y is defined as:
  where f(x) and g(y) are the probability density functions of X and Y respectively.
  One may easily show that such metrics above do not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space. In fact they satisfy this condition if and only if both arguments X, Y are certain events described by Dirac delta density probability distribution functions. In such a case:
  the Łukaszyk–Karmowski metric simply transforms into the metric between expected values ,  of the variables X and Y and obviously:
  For all the other cases however:
  The Łukaszyk–Karmowski metric satisfies the remaining non-negativity and symmetry conditions of metric directly from its definition (symmetry of modulus), as well as subadditivity/triangle inequality condition:
  Thus
  In the case where X and Y are dependent on each other, having a joint probability density function f(x, y), the L–K metric has the following form:
caption: L–K metric between two random variables X and Y having normal distributions and the same standard deviation 
    
      
        
          σ
          =
          0
          ,
          σ
          =
          0.2
          ,
          σ
          =
          0.4
          ,
          σ
          =
          0.6
          ,
          σ
          =
          0.8
          ,
          σ
          =
          1
        
      
      {\displaystyle \sigma =0,\sigma =0.2,\sigma =0.4,\sigma =0.6,\sigma =0.8,\sigma =1}
    
   (starting with the bottom curve). 
    
      
        
          
            m
            
              x
              y
            
          
          =
          
            |
          
          
            μ
            
              x
            
          
          −
          
            μ
            
              y
            
          
          
            |
          
        
      
      {\displaystyle m_{xy}=|\mu _{x}-\mu _{y}|}
    
   denotes a distance between means of X and Y.
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