# Math Help - Expected Value and Variance (Basic Properties)

1. ## Expected Value and Variance (Basic Properties)

Hey guys, Im going over my Econometrics homework () and was hoping for some help to confirm or correct my answers. Its regarding the basic properties of Variance and Expected Value. Any help is much appreciated. Here's what I got so far (my answers in italicized font):

1) Calculate E(X + Y) = muX + muY
2) Calculate V(X + Y) = sigmaX + sigmaY + 2sigmaXY
3) Calculate E(2X) = 2muX
4) Calculate E(2Y) = ???
5) Calculate V(2X + 3Y) = 4sigmaX + 9sigmaY + 12sigmaXY
6) Calculate E(1 + 2X) = 2muX + 1
7) Calculate V(1 + 2X) = 4sigmaX
8) Calculate E(3X + 4(Y + 1)) = 3muX + 4muY + 4
9) Calculate V(3X + 4(Y + 1)) = 9sigmaX + 16sigmaY
10) Calculate E(aX + b(Y + c)) = amuX + bmuY + bc
11) Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY

2. Hello,
Originally Posted by RutgersGirl
Hey guys, Im going over my Econometrics homework () and was hoping for some help to confirm or correct my answers. Its regarding the basic properties of Variance and Expected Value. Any help is much appreciated. Here's what I got so far (my answers in italicized font):
Does sigma stand for the variance ? Because all the sigma's you've used should be variances
And usually, sigma stands for standard deviation.

1) Calculate E(X + Y) = muX + muY

2) Calculate V(X + Y) = sigmaX + sigmaY + 2sigmaXY
No, the formula is V(X+Y)=V(X)+V(Y)+2Cov(X,Y)
where Cov denotes the covariance. Note that Cov(X,Y)=0 if X and Y are independent.
Cov(X,Y)=E[(X-mu(X))(Y-mu(Y))=E(XY)-E(X)E(Y) (the red one is the most commonly used)
3) Calculate E(2X) = 2muX

4) Calculate E(2Y) = ???
Exactly the same as above in 3)
For any random variable X (or Y, or whater your variable is called), E(aX)=aE(X)
5) Calculate V(2X + 3Y) = 4sigmaX + 9sigmaY + 12sigmaXY
There is still a problem with the 12sigma(XY), otherwise, it's correct.
6) Calculate E(1 + 2X) = 2muX + 1

7) Calculate V(1 + 2X) = 4sigmaX

8) Calculate E(3X + 4(Y + 1)) = 3muX + 4muY + 4

9) Calculate V(3X + 4(Y + 1)) = 9sigmaX + 16sigmaY
Once again, look at the covariance. Don't hesitate to ask if you don't know what to do.
10) Calculate E(aX + b(Y + c)) = amuX + bmuY + bc

11) Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY
Once again, the formula for the variance is false

Good working anyway, it's good to see someone's attempts :P

3. Wow! Thanks for all your help. I was able to get the homework down, but I didn't do as good as I was hoping on the exam. Much appreciated!

4. Her sigma_xy IS covariance in most books.
So that's just notation.
The only thing I see wrong is number 11.
You're missing the covariance term.
Calculate V(aX + b(Y + c)) = a^2sigmaX + b^2sigmaY
you need to add 2ab times the covariance of X and Y.
The constant term bc disappears when you calculate variance.
So this is the same as V(aX + bY),
since
aX + b(Y + c) = aX + bY + bc

5. Originally Posted by matheagle
Her sigma_xy IS covariance in most books.
So that's just notation.
The only thing I see wrong is number 11.
You're missing the covariance term.
But $\sigma_X$ does stand for the standard deviation. And all the $\sigma_X$ and $\sigma_Y$ should be variances.