I am learning probability theory by myself and have a problem with independence of two random variables:
Suppose that I have $X_1, X_2, Y_1, Y_2$ are i.i.d N(0,1), then I define:
$X=X_1+X_2, Y=Y_1+Y_2$. Intuitively, X and Y are independent and I am trying to prove that by showing:
$F_{X,Y}(x,y)=F_X(x)F_Y(y)$ or $f_{X,Y}(x,y)=f_X(x)f_Y(y)$
I found $f_X(x)= \frac{1}{\sqrt{4\pi}} exp({-\frac{x^2}{4}})$ and $f_Y(y)= \frac{1}{\sqrt{4\pi}} exp({-\frac{y^2}{4}})$. However, I cannot find $f_{XY}(x,y)$ and thus, I do not know how to show X and Y are independent. I wonder if I approach the problem in the right way??
Many thanks.
For every independent random variables $X_1$, $X_2$, $Y_1$, $Y_2$, and for every measurable functions $\xi$ and $\eta$, the random variables $X=\xi(X_1,X_2)$ and $Y=\eta(Y_1,Y_2)$ are independent.