1.6Means--Moments--Correlations for Multidimensional Distributions I

Given n-dimensional distribution :

• The mean now becomes an n-dimensional vector

  

• The variance becomes an moment matrix M

  

• Note for we can show---analogously to Equation (9)---that

  

• Also note that M is symmetric and positive semi-definite. [This follows because ---any y---clearly has a integrand].

• Now the diagonal terms of M are called variances which are :

• Their square roots are standard deviations again.

• Further, when , we put , which is and is called the correlation coefficient.