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An iid random sample of 4 is taken from a normal distribution with mean 2 and variance 3. What is the covariance matrix? What is the matrix of mew? If they are not iid, then how would the covariance matrix differ?

My solution:

If it is iid, the matrix is simply a diagonal matrix with 3 as its entries

The mean matrix is just a row matrix with entries 2

If it is not iid, the matrix has diagonals of 3 and non-diagonal entries I am not sure of...

Please let me know if my solution is correct and how to find the matrix if they are not iid.

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That's correct. If they are not independent (but each individually still has this same distribution), all you know about the covariance matrix is that it is a $4 \times 4$ positive semidefinite matrix with diagonal elements $3$. An off-diagonal element could be anything from $-3$ to $3$.

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  • $\begingroup$ No, there are stricter limits on how close to -3 the covariances can be. In the equicorrelation case, the lower limit is $-3\frac1{4-1}$ $\endgroup$ – kjetil b halvorsen Dec 18 '17 at 10:08
  • $\begingroup$ If $X_2$ is $2\mu-X_1$, the covariance of $X_1$ and $X_2$ is $-3$. $\endgroup$ – Robert Israel Dec 18 '17 at 19:31

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