Expected value and variance for random vectors

Problem

\(X\) is \(2×1\) random vector, \(X=\left( X_1,X_2 \right)\) with expected value \(μ={\left( 2,-2 \right)}'\) and variance

\[Var\left( X \right)=\begin{bmatrix}1 & 0.5 \\ 0.5 & 1\end{bmatrix}\]

\(Y=X_1+X_2\) .

1. Find \(a\) such that \(Y=a'X\)
2. Find \(E\left( Y \right)\) using the result, \(E\left( Y \right)=a'μ\)
3. Find \(Var\left( Y \right)\) using the result, \(Var\left( Y \right)=a'Var\left( X \right)a\)
4. Double check c) using \(Var\left( Y \right)=Var\left( X_1+X_2 \right)=Var\left( X_1 \right)+Var\left( X_2 \right)+2Cov\left( X_1,X_2 \right)\)