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...
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 $\displaystyle Var(X)= E[(X-\mu_x)^2] $.
After expanding and eliminating you will get $\displaystyle Var(X) =E(X^2)-(E(X))^2$
For two variable, you substiute X with XY, it becomes
$\displaystyle Var(XY)=E(X^2Y^2)-(E(XY))^2 $
$\displaystyle =E(X^2)E(Y^2)-(E(X))^2(E(Y^2))^2$
$\displaystyle = Var(X)E(Y)+Var(Y)E(X)+Var(X)Var(Y)$----eq(1)
where $\displaystyle Var(X) = E(X^2)-(E(X))^2$ and $\displaystyle Var(Y) = E(Y^2)-(E(Y))^2$--------------------eq(2)
When X and Y are independent $\displaystyle E(XY)=\int_x xf(x)[\int_y yg(y)dy]dx=\int_x xf(x)\int_y E(Y)dx=E(X)E(Y)$------eq(3)
Substitute eq(2) and eq(3) into eq(1). You should get the final form you find in you book.