Let $f(x,y,z)=x^2e^{-x-xy-xz}$, if $x,y,z>0$ and $f(x,y,z)=0$ otherwise. Are the continuous random variables $x,y,z$ independent or not?
Intuitively they are not independent. I calculated the marginal density functions:
$f(x)=\iint f(x,y,z) dydz=e^{-x}$
$f(y)=\iint f(x,y,z) dxdz=(y+1)^{-2}$
$f(z)=\iint f(x,y,z) dxdy=(z+1)^{-2}$
Now we observe that if $x,y,z>0$,
$f(x,y,z)=x^2e^{-x-xy-xz}\neq{e^{-x}}(y+1)^{-2}(z+1)^{-2}$. Thus $x,y,z$ are not independent. Is this correct? Is there easier method to check if they are independent as this way is a bit tedious?