i am trying to refresh my knowledge of the multivariate normal distribution. the standard formula as per belowenter image description here

i would normally think of x as a tall and slim matrix of the covariate values (rows representing different observations (e.g. 1:N), columns being the different covariates (e.g. 1:D)). I am hence imagining the transpose (x-mu)^T as a a short and wide matrix - e.g. for 2 covariates and 6 observations something like

     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    1    2    3    4    5    6
[2,]    7    8    9   10   11   12

However - clearly this would not commute / multiply with the correlation matrix Sigma - which would be e.g. 2*2 in this specific example case. what is the thinking error that i am making? any tips much appreciated


$x$ is a column vector, so $(x-\mu)^{T}$ is a $1$ by $n$ row matrix. $\Sigma^{-1}$ is of size $n$ by $n$. Here $n$ is the length of the random vector $X$. There isn't any "data" in this probability density function.


No, in this case x is a vector representing the result of sampling from the distribution and $\mu$ is a vector of the same size. Each element within the $\mu$ vector represents the mean of the distribution along that dimension; thus, $(x-\mu)^T$ if also a vector, but is simply transposed so it becomes a row vector instead of a column vector.


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