1. ## mean and variance

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

2. Hello,

Originally Posted by haleyGeorge
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
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 $\displaystyle var(2X_1-3X_2+4X_3)=4var(X_1)+9var(X_2)+16var(X_3)$

3. I know this is probably a really stupid question, but do I just plug in the numbers given for mean and variance into the equation? Or what is my next step?

4. Originally Posted by haleyGeorge
I know this is probably a really stupid question, but do I just plug in the numbers given for mean and variance into the equation? Or what is my next step?
Yes, you just plug them in it

5. ok, then for the first one I got E(Y) = -7 and var(Y) = 155. Is that correct? Thanks for all your help!

6. Originally Posted by haleyGeorge
ok, then for the first one I got E(Y) = -7 and var(Y) = 155. Is that correct? Thanks for all your help!
Yes !!!

How much do you find for Z ?

7. E(Z) = -17 and var(Z) = 36