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Math Help - Issues With Expectations

  1. #1
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    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.

    I found the mean. I just need help finding Var(Z). Please help me out. Thanks.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by helium0204 View Post
    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.

    I found the mean. I just need help finding Var(Z). Please help me out. Thanks.
    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 ???
    Last edited by Moo; August 3rd 2008 at 01:09 PM.
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  3. #3
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    Quote Originally Posted by Moo View Post
    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
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  4. #4
    Moo
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    Quote Originally Posted by helium0204 View Post
    p is correlation coefficient
    Okay

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

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