If X1, X2, and X3 are independent and have the means 4, 9, and 3 and the vaiance 3,7, and 5, find the mean and the variance of:
a. Y = 2X1 -3X2 +4X3
b. Z = X1 -2X2 -X3
Hello,
The mean is a linear application.
Therefore : $\displaystyle E(aX)=aE(X)$ and $\displaystyle E(X+Y)=E(X)+E(Y)$, E designing the mean.
From the definition, $\displaystyle var(X+Y)=var(X)+var(Y)+2cov(X,Y)$
But if X and Y are independent, then $\displaystyle cov(X,Y)=0 \implies var(X+Y)=var(X)+var(Y)$
Plus, $\displaystyle var(X)=E(X^2)-(E(X))^2 \implies var(aX)=E((aX)^2)-(E(aX))^2=a^2 var(X)$
From there, for the first one for example :
$\displaystyle E(2X_1-3X_2+4X_3)=2E(X_1)-3E(X_2)+4E(X_3)$
And $\displaystyle var(2X_1-3X_2+4X_3)=4var(X_1)+9var(X_2)+16var(X_3)$