Variance of the product of two independent variables

**paulrb** · December 17th 2009, 09:05 AM

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...

**novice** · December 17th 2009, 11:20 AM

Originally Posted by paulrb

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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