stats_linear_combo p2 Y=aX+b,L$

$Y=aX+b,L$

$$\begin{array}{c}\\ Y=3X+4\\ X\sim N(\mu =2,\sigma =5)\\ L=2H_1-3H_2+1H_3\\ H_i{\sim }^{iid}N(\mu =2,{\sigma }^2=3)\end{array}$$

Finding E(#) plug mu in, V(#) if iid directly plug ${\sigma }^2$ into entire expr

E(Y)

$$3\ast 2+4=10$$

E(L)

$$\begin{array}{c}(2-3+1)\ast 2\\ [1]0\end{array}$$

Problem	Formula (the first line)	Information Used (mu)	Calculation	Answer
E(Y)	$E(3X+4)=3E(X)+4$	$E(X)=2$	$3(2)+4=6+4$	10
E(L)	$E(2H_1-3H_2+H_3)=2E(H_1)-3E(H_2)+E(H_3)$	$E(H_i)=2$	$2(2)-3(2)+1(2)=4-6+2$	0

V(y)

$$3^2\ast 5^2=9\ast 25=225$$

V(L)

$$\begin{array}{c}(4+9+1)\ast 3\\ [1]42\end{array}$$

Problem	Formula	Information Used (sigma)	Calculation	Answer
V(Y)	$V(3X+4)=3^{2}V(X)$	$V(X)=5^2=25$ (drop +b term)	$9\cdot 25$	225
V(L)	$V(2H_1-3H_2+H_3)=2^{2}V(H_1)+(-3{)}^{2}V(H_2)+1^{2}V(H_3)$	$V(H_i)=3$ (and independence ${\sim }^{\text{iid}}$)	$4(3)+9(3)+1(3)=12+27+3$	42