For each of the joint PDFs listed below, determined which random variables are independent and which are not.

a. \(f(x, y)=e^{-(x+y)} I_{[0, \infty)}(x) I_{[0, \infty)}(y)\)

b. \(f(x, y)=\frac{x(1+y)}{300} I_{\{1,2,3,4,5\}}(x) I_{\{1,2,3,4,5\}}(y)\)

c. \(f(x, y, z)=8 x y z I_{[0,1]}(x) I_{[0,1]}(y) I_{[0,1]}(z)\)

d. \(f\left(x_{1}, x_{2}, x_{3}ight)=\frac{.5^{x_{1}} \cdot 2^{x_{2}} \cdot 75^{x_{3}}}{10} \prod_{i=1}^{3} I_{\{0,1,2, \ldots\}}\left(x_{i}ight)\)