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Suppose we have 4 normally distributed variables X1, X2, X3 and X4. The covariance matrix and mean values of X1 and X2 are given. Is there a way to determine the mean values of variables X3 and X4?

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No, there is no way to estimate the unknown means in general. – Dilip Sarwate Apr 7 '12 at 23:11
The covariances are based on the centered random variables X-E(X), hence they cannot feel the expectations E(X). – Did Apr 7 '12 at 23:21
I forgot to mention the fact that the variables are all normally distributed, but I guess that doesn't change much. – Jacques Apr 8 '12 at 6:06

No, you know nothing about the means of $X_3$ and $X_4$.

Suppose you let $Y_3 = X_3 + k_3$ and $Y_4 = X_4 + k_4$ for real constants $k_3$ and $k_4$.

Then looking at $X_1$, $X_2$, $Y_3$ and $Y_4$: the means of $X_1$ and $X_2$ would still be the same, and the covariance matrix would also be the same, even though the means of the third and fourth variables had changed by arbitrary amounts.

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