# Issues With Expectations

• Aug 3rd 2008, 08:05 AM
helium0204
Issues With Expectations
Here's the problem that I am having.

Let X and Y be random variables with E(X)=1, E(Y)= 4 Var(X)= 4 Var(Y)=6 p=1/2. Find mean and variance of Z= 3X-2Y.

• Aug 3rd 2008, 12:57 PM
Moo
Hello,
Quote:

Originally Posted by helium0204
Here's the problem that I am having.

Let X and Y be random variables with E(X)=1, E(Y)= 4 Var(X)= 4 Var(Y)=6 p=1/2. Find mean and variance of Z= 3X-2Y.

For this, you have to remember that the mean is a linear function, that is to say E(aX+bY)=aE(X)+bE(Y)

For the variance, the formula is : var(aX+bY)=a^2 var(X)+b^2 var(Y), assuming that X and Y are independent. Otherwise you will need more information.
You can prove this formula with the definition of the variance that includes the mean :
var(X)=E(X^2)
and var(X+Y)=var(X)+var(Y) if X and Y are independent.

Edit : What is p ???
• Aug 3rd 2008, 01:52 PM
helium0204
Quote:

Originally Posted by Moo
Hello,

For this, you have to remember that the mean is a linear function, that is to say E(aX+bY)=aE(X)+bE(Y)

For the variance, the formula is : var(aX+bY)=a^2 var(X)+b^2 var(Y), assuming that X and Y are independent. Otherwise you will need more information.
You can prove this formula with the definition of the variance that includes the mean :
var(X)=E(X^2)
and var(X+Y)=var(X)+var(Y) if X and Y are independent.

Edit : What is p ???

p is correlation coefficient
• Aug 3rd 2008, 03:48 PM
Moo
Quote:

Originally Posted by helium0204
p is correlation coefficient

Okay

Use the fact that the correlation coefficient is equal to :
$\displaystyle \frac{\text{covariance}(X,Y)}{\sigma_X \cdot \sigma_Y}$, where $\displaystyle \sigma$ represents the standard deviation, that is to say $\displaystyle \sqrt{\text{variance}}$.

Then, use the following formula : var(X+Y)=var(X)+var(Y)+2 cov(X,Y), where cov is the covariance