Graphical display of joint probability function
The probability function of a single discrete random variables can be displayed graphically in a 2-dimensional bar chart. Although of less practical importance, the joint probability function of two discrete variables can be displayed in a similar 3-dimensional bar chart.
Maximum and minimum of three rolled dice
Consider rolls of three fair dice for which there is probability \(\frac 1 6\) for each value. We define \(X\) and \(Y\) to be the maximum and minimum of the three values. We state here without proof the joint probability function of these two variables.
\[ p(x,y) \;\;=\;\; \begin{cases} {\frac 1 {6^3}} & \quad\text{if }x = y \;\;\text{ and }\;\; 1 \le x,y \le 6 \\[0.4em] {\frac {x-y}{6^2}} & \quad\text{if } 1 \le y \lt x \le 6 \\[0.4em] 0 & \quad\text{otherwise} \end{cases} \]The diagram below shows a 3-dimensional bar chart of these probabilities.
Drag the centre of the diagram to rotate the diagram and get a better feel for how the probabilities vary.
Observe that the most likely combination is for a minimum of 1 and a maximum of 6. (This allows more possibilities for the third value.)