Determine standard deviation of $Y = 3X_1 - 2X_2 + X_3 - 6$

Question

$X_1, X_2, X_3$ are independent stochastic variables, all with expected value 2 and standard deviation 3.

With $$Y = 3X_1 - 2X_2 + X_3 - 6,$$

determine $D(Y)$.

I know that $D(Y) = \sqrt{V(X)}$, and I know that $V(Y) = E(Y^2) - E(Y)^2$, however I don't know how to calculate $E(Y^2)$ or for that matter $V(Y)$ directly.

Answer 1

$\sigma_Y^2 = 9\sigma_{X_{1}}^2+ 4\sigma_{X_{2}}^2 + \sigma_{X_{3}}^2 -12\sigma_{X_1}\sigma_{X_2}\rho_{X_1,X_2} + 6\sigma_{X_1}\sigma_{X_3}\rho_{X_1,X_3} - 4\sigma_{X_2}\sigma_{X_3}\rho_{X_2,X_3}$

if $X_1, X_2, X_3$ are independent than all of the correlation terms equal $0$

$\sigma_Y^2 = 9\sigma_{X_{1}}^2+ 4\sigma_{X_{2}}^2 + \sigma_{X_{3}}^2$