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I'm trying to wrap my head around Gaussian random vectors, in particular their covariance matrices. I've seen two different definitions of a covariance matrix depending on whether you are calculating the matrix from observed results or from a mathematical model. If I let a Gaussian random vector be defined as such. $$X = \mu + AZ$$ Where $\mu \in \mathbb{R}^n$, $A \in M_{n,k}$ and $Z = \{Z_0, ..., Z_k\}^T$ where $Z_i$ are independent and identically distributed standard normal random variables.

I've seen that one definition of the covariance matrix $\Sigma$ is $AA^T$. And another is $\Sigma_{i,j} = Cov[X_i, X_j] = E[(X_i-\mu_i)(X_j-\mu_j)]$. I attempted to check that these are equivalent with an example. $$X = \begin{align}\begin{bmatrix}1 \\ 2\end{bmatrix} + \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}\begin{bmatrix}Z_0 \\ Z_1\end{bmatrix}\end{align}$$ I found, $$\Sigma = AA^T = \begin{align}\begin{bmatrix}5 & 11 \\ 11 & 25\end{bmatrix}\end{align}$$ And, $$\Sigma_{0,1} = E[(X_0- \mu_0)(X_1 - \mu_1)] = E[(Z_0 + 2Z_1)(3Z_0+4Z_1)] = E[3Z_0^2 + 10Z_0Z_1 + 8Z_1^2]$$ Now I have no inclination as to if this makes any sense or why it might be true but I saw that if I let $E[Z_iZ_j] = \delta_{i,j}$ (Kronecker delta). Then we successfully find $\Sigma_{0,1} = 11$.

Could someone please provide more clarity on what is actually happening here?

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    $\begingroup$ $E[Z_iZ_j] = \delta_{ij}$ is because $Z_0, Z_1$ are i.i.d. $N(0, 1)$. $\endgroup$
    – Zhanxiong
    Commented Feb 16, 2023 at 21:51
  • $\begingroup$ @Zhanxiong Ok, I think I understand why when $i \neq j$ the expectation is zero (both have mean value 0). And I understand that when $i = j$ the expected value would be positive. But it's exactly 1? Why? $\endgroup$ Commented Feb 16, 2023 at 22:11
  • $\begingroup$ @repanda2236: Do you know if $Z \sim N(0, 1)$, then $E[Z^2] = 1$? $\endgroup$
    – Zhanxiong
    Commented Feb 16, 2023 at 22:17
  • $\begingroup$ @Zhanxiong No that's new to me, I'm trying to look up a proof but I can't seem to get the right search terms to find anything related. $\endgroup$ Commented Feb 16, 2023 at 22:21
  • $\begingroup$ @repanda2236: By definition, if $Z \sim N(0, 1)$, then $E[Z] = 0, \operatorname{Var}(Z) = 1$, it then follows that $E[Z^2] = \operatorname{Var}(Z) + (E[Z])^2 = 1$. I am pretty sure if you checked the Wikipedia page of "normal distribution", you will find all these facts. $\endgroup$
    – Zhanxiong
    Commented Feb 16, 2023 at 22:47

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