page_title: Multivariate normal distribution
hierarchical_section_title: Multivariate normal distribution
caption_attribution_description: English: Illustration of a multivariate gaussian distribution and its marginals. Matlab code provided below.
context_page_description: In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of correlated real-valued random variables each of which clusters around a mean value.
context_section_description: In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions.  One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.
caption: Many sample points from a multivariate normal distribution with 
    
      
        
          
            μ
          
          =
          
            [
            
              
                
                  
                    
                      0
                    
                  
                  
                    
                      0
                    
                  
                
              
            
            ]
          
        
      
      {\displaystyle {\boldsymbol {\mu }}=\left[{\begin{smallmatrix}0\\0\end{smallmatrix}}\right]}
    
   and 
    
      
        
          
            Σ
          
          =
          
            [
            
              
                
                  
                    
                      1
                    
                    
                      3
                      
                        /
                      
                      5
                    
                  
                  
                    
                      3
                      
                        /
                      
                      5
                    
                    
                      2
                    
                  
                
              
            
            ]
          
        
      
      {\displaystyle {\boldsymbol {\Sigma }}=\left[{\begin{smallmatrix}1&3/5\\3/5&2\end{smallmatrix}}\right]}
    
  , shown along with the 3-sigma ellipse, the two marginal distributions, and the two 1-d histograms.
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