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.


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 $3$.

  • $\begingroup$ 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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