E: 6, V:43
Suppose $X$ and $Y$ are independent with $E[X]=1$, $E[Y]=2$, $V[X]=3$, and $V[Y]=1$. Find the mean and variance of $3X+4Y-5$.
E: 6, V:43
This is a straightforward application of properties of variance, expectation, and independence.
$$\begin{aligned}E[3X + 4Y - 5] &= E[3X] + E[4Y] - E[5] \&= 3E[X] + 4E[Y] - 5 \&= 3(1) + 4(2)- 5 \&= 3 + 8 - 5 \&= \mathbf{6}\end{aligned}$$
$$\begin{aligned}V[3X+4Y-5] &= V[3X] + V[4Y] + V[5] + 2Cov[3X,4Y] \&= 9V[X] + 16V[Y] + 0 + 0 \&= 9(3) + 16(1) \&= 27 + 16 \&= \mathbf{43}\end{aligned}$$