The two dice have PMFs
[tex]P(X=x)=\begin{cases}\dfrac12&\text{for }x\in\{0,2\}\\\\0&\text{otherwise}\end{cases}[/tex]
[tex]P(Y=y)=\begin{cases}\dfrac14&\text{for }y\in\{0,1,4,5\}\\\\0&\text{otherwise}\end{cases}[/tex]
The sum of the two rolls can take on 8 different values:
[tex]W=\begin{cases}0&\text{if }x=0,y=0\\1&\text{if }x=0,y=1\\2&\text{if }x=2,y=0\\3&\text{if }x=2,y=1\\4&\text{if }x=0,y=4\\5&\text{if }x=0,y=5\\6&\text{if }x=2,y=4\\7&\text{if }x=2,y=5\end{cases}[/tex]
The outcome of either roll is independent of the other, so that
[tex]P(X=x\text{ and }Y=y)=P(X=x)P(Y=y)[/tex]
a. Then PMF of [tex]W[/tex] is
[tex]P(W=w)=\begin{cases}\dfrac12\cdot\dfrac14=\dfrac18&\text{for }w\in\{0,1,2,3,4,5,6,7\}\\\\0&\text{otherwise}\end{cases}[/tex]
b. The histogram is nothing special, same as the discrete uniform distribution over the interval [0, 7].
