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.

Chelsea Pruitt 2022-10-23 Answered
Normal n-distribution and Variance-covariance matrix
Let x be a standard normal vector of length n.<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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Answers (1)

Zackary Mack
Answered 2022-10-24 Author has 12 answers
Step 1
We 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 2
We usually specify the vector of means of each component and the covariance matrix, whose (i,j) entry is Cov ( Y i , Y j )
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