Given E(X) , V(X) , E(Y) , V(Y) , COV(X,Y)
As we know :
Expectation follows linearity of expression (meaning each term can be separated)
So here : E(X + 2Y) = E(X) + E(2Y)
= E(X) + 2 E(Y) [ Follows from the simple fact : If we multiply each element of Y by 2 , then expectation(mean) is also going to be multiplied by 2 ].
Now : for Var(X - 2Y + 1) , we need to know :
Variance only gets affected by scale factor e.g. in -2Y , scale factor is -2 and not by constant terms i.e. '1' here..In other words constant terms do not contribute to variance.
So general formula for Var(aX + bY + c) = a2 Var(X) + b2 Var(Y) + 2ab Cov(X,Y) where Cov(X,Y) is covariance of X and Y[Var(X) is multiplied by a2 because standard deviation will scale by |a| and hence variance which is square of standard deviation will be squared as well ]
Hence Var(X - 2Y + 1) = 12 Var(X) + (-2)2 Var(Y) + 2(1)(-2) Cov(X,Y)
= Var(X) + 4 Var(Y) - 4 Cov(X,Y)
Cov(X,Y) is found as : E(X.Y) - E(X).E(Y) . Cov(X,X) is nothing but Var(X)..