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Math Help - Variance of the product of two independent variables

  1. #1
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    Variance of the product of two independent variables

    How can I get Var(XY) when X,Y are independent?

    I am mainly interested in the method of getting a formula for this than the formula itself, because my book has a lot of these formulas but I don't want to blindly memorize them...
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  2. #2
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    Quote Originally Posted by paulrb View Post
    How can I get Var(XY) when X,Y are independent?

    I am mainly interested in the method of getting a formula for this than the formula itself, because my book has a lot of these formulas but I don't want to blindly memorize them...
    I will show you the process, but you need to do the rest.

    The definition of variance with a single random variable is Var(X)= E[(X-\mu_x)^2] .

    After expanding and eliminating you will get Var(X) =E(X^2)-(E(X))^2

    For two variable, you substiute X with XY, it becomes

    Var(XY)=E(X^2Y^2)-(E(XY))^2
    =E(X^2)E(Y^2)-(E(X))^2(E(Y^2))^2
    = Var(X)E(Y)+Var(Y)E(X)+Var(X)Var(Y)----eq(1)

    where Var(X) = E(X^2)-(E(X))^2 and Var(Y) = E(Y^2)-(E(Y))^2--------------------eq(2)

    When X and Y are independent  E(XY)=\int_x xf(x)[\int_y yg(y)dy]dx=\int_x xf(x)\int_y E(Y)dx=E(X)E(Y)------eq(3)

    Substitute eq(2) and eq(3) into eq(1). You should get the final form you find in you book.
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