- #1

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Intuitively they are not independent. I calculated the marginal density functions:

##f_x(x)=\iint_{\Omega} f(x,y,z) dydz=e^{-x}##

##f_y(y)=\iint_{\Omega} f(x,y,z) dxdz=(y+1)^{-2}##

##f_z(z)=\iint_{\Omega} f(x,y,z) dxdy=(z+1)^{-2}##

Now we observe that if ##x,y,z>0##,

##(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'?