Expected value from covariances and variances

Question

I am given three r.v $X_1, X_2, X_3$, $Cov(X_1,X_2), Cov(X_2, X_3), Cov(X_1,X_3)$ and $Var(X_1), Var(X_2), Var(X_3)$. From the given, is it possible to find $E(Y)$ where $Y = a_1X_1 + a_2X_2 + a_3X_3$?

Answer 1

No, that is not possible without further information.

If for $i=1,2,3$ we have $X_i'=X_i+p_i$ then we find exactly the same covariances and variances in the sense that: $$\mathsf{Cov}(X_i',X_j')=\mathsf{Cov}(X_i,X_j)$$for $i,j=1,2,3$.

However this with: $$Y'=a_1X_1'+a_2X_2'+a_3X_3'=Y+a_1p_1+a_2p_2+a_3p_3$$ so that: $$\mathbb EY'=\mathbb EY+a_1p_1+a_2p_2+a_3p_3$$