Draw a histogram for its probability function and a graph for its distribution function. [closed]

Question

We roll two fair dice, a blue and a red one, independently of each other. Let X be the number obtained on the blue die minus the number obtained on the red die. Draw a histogram for its probability function and a graph for its distribution function.

Answer 1

Hint: The sample space for $X=\color{blue}{B}-\color{red}{R}$ decomposes as the cartesian product $\{(\color{blue}{B},\color{red}{R})\}=\{1,2,\dots,6\}^2$ in terms of the values obtained on each colored die (red & blue respectively). $$ \begin{array}{rrrrrrr} X && \color{red}{1} & \color{red}{2} & \color{red}{3} & \color{red}{4} & \color{red}{5} & \color{red}{6} \\\\ \color{blue}{1} && 0 & -1 & -2 & -3 & -4 & -5 \\ \color{blue}{2} && 1 & 0 & -1 & -2 & -3 & -4 \\ \color{blue}{3} && 2 & 1 & 0 & -1 & -2 & -3 \\ \color{blue}{4} && 3 & 2 & 1 & 0 & -1 & -2 \\ \color{blue}{5} && 4 & 3 & 2 & 1 & 0 & -1 \\ \color{blue}{6} && 5 & 4 & 3 & 2 & 1 & 0 \end{array} $$ Can you deduce the histogram from this and then normalize that to get the probability density function?