Question: The joint probability mass function of two discrete variables X and Y is given by

\[\begin{aligned} & f\left( x,y \right)=\left\{ \begin{aligned} & c\left( 3x+y \right)\,\,\,\,\,\,\,\,\,x=0,1,2,\,\,\,y=2,3 \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \\ \end{aligned} \right. \\ & \\ \end{aligned}\]

(a) Find c.

(b) Find the marginal probability functions

(c) Compute \(E\left( X \right),\operatorname{var}\left( X \right)\)

(d) Are X and Y independent.