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Math Help - mean and variance

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    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
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    Hello,

    Quote Originally Posted by haleyGeorge View Post
    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 : E(aX)=aE(X) and E(X+Y)=E(X)+E(Y), E designing the mean.


    From the definition, var(X+Y)=var(X)+var(Y)+2cov(X,Y)
    But if X and Y are independent, then cov(X,Y)=0 \implies var(X+Y)=var(X)+var(Y)

    Plus, 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 :

    E(2X_1-3X_2+4X_3)=2E(X_1)-3E(X_2)+4E(X_3)

    And var(2X_1-3X_2+4X_3)=4var(X_1)+9var(X_2)+16var(X_3)
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    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?
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    Quote Originally Posted by haleyGeorge View Post
    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
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    ok, then for the first one I got E(Y) = -7 and var(Y) = 155. Is that correct? Thanks for all your help!
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    Quote Originally Posted by haleyGeorge View Post
    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 ?
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    E(Z) = -17 and var(Z) = 36
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