Normal n-distribution and Variance-covariance matrixLet x be a standard normal vector of length n.&lt;br.Let A ∈ R m × n be a matrix.Then we say that Ax has normal distributed components. Or we could say that Ax is a vector whose components are normally distributed with length m.Example: A = [ 3 − 2 2 1 ] x = ( d ) N ( O , I 2 × 2 ) = [ x y ] A x = [ 3 X − 2 Y 2 X + Y ] 3 X − 2 Y = ( d ) N ( 0 , 3 2 + 2 2 ) = N ( 0 , 13 ) 2 X + Y = ( d ) N ( 0 , 4 2 + 1 ) = N ( 0 , 5 )

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Normal n-distribution and Variance-covariance matrixLet x be a standard normal vector of length n.&amp;lt;br.Let   A  ∈            R              m      ×      n       be a matrix.Then we say that Ax has normal distributed components. Or we could say that Ax is a vector whose components are normally distributed with length m.Example:  A  =      [                            3                          −          2                                      2                          1                      ]      x      =          (      d      )        N  (  O  ,      I          2      ×      2        )  =      [                            x                                      y                      ]      A  x  =      [                            3          X          −          2          Y                                      2          X          +          Y                      ]    3  X  −  2  Y      =          (      d      )        N  (  0  ,      3    2    +      2    2    )  =  N  (  0  ,  13  )    2  X  +  Y      =          (      d      )        N  (  0  ,      4    2    +  1  )  =  N  (  0  ,  5  )

Zackary Mack · Accepted Answer

Step 1We usually say that Ax is a Gaussian vector of length n, that is, for each       c    1    ,  …  ,      c    n    ∈      R  , denoting       Y    j   the j-th component of Ax, the random variable       ∑          j      =      1        n        c    j        Y    j   has a Gaussian (possibly degenerated) distribution.Step 2We usually specify the vector of means of each component and the covariance matrix, whose (i,j) entry is   Cov  ⁡  (      Y    i    ,      Y    j    )

Normal n-distribution and Variance-covariance matrix. Let x be a standard normal vector of length n. Let A in R^{m times n} be a matrix.

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