Definition
The marginal probability function of \(X\) is
\[ p_X(x) \;=\; P(X = x) \;=\; \sum_{y} p(x,y) \]In the same way, the marginal probability function of \(Y\) is
\[ p_Y(y) \;=\; P(Y = y) \;=\; \sum_{x} p(x,y) \]These describe the distributions of the separate variables when nothing is known about the value of the other variable.
Maximum and minimum of three dice
On the previous page, we gave the joint probability function for the maximum, \(X\), and minimum, \(Y\), of three independent rolls of a fair 6-sided die.
\[ p(x,y) \;\;=\;\; \begin{cases} {\frac 1 {6^3}} & \quad\text{if }x = y \;\;\text{ and }\;\; 1 \le x,y \le 6 \\[0.4em] {\frac {x-y}{6^2}} & \quad\text{if }y \lt x \text{, }\;\; y \ge 1 \;\;\text{ and }\;\; x \le 6 \\[0.4em] 0 & \quad\text{otherwise} \end{cases} \]The marginal probability function for \(X\) can be found by adding the joint probabilities over \(Y\).
\[ \begin{align} p_X(6) \;&=\; p(6,6) + p(6,5) + \cdots + p(6,1) \\ &=\; \frac 1{6^3} + \frac 1{6^2} + \frac 2{6^2} + \cdots + \frac 5{6^2} \\ &=\; \frac 1{6^3} + \frac {1+2+3+4+5}{6^2} \\ &=\; \frac 1{6^3} + \frac {15}{6^2} \\[0.7em] p_X(5) \;&=\; p(5,5) + p(5,4) + \cdots + p(5,1) \\ &=\; \frac 1{6^3} + \frac 1{6^2} + \frac 2{6^2} + \cdots + \frac 4{6^2} \\ &=\; \frac 1{6^3} + \frac {10}{6^2} \\[0.7em] p_X(4) \;&=\; \frac 1{6^3} + \frac {6}{6^2} \\[0.7em] p_X(3) \;&=\; \frac 1{6^3} + \frac {3}{6^2} \\[0.7em] p_X(2) \;&=\; \frac 1{6^3} + \frac {1}{6^2} \\[0.7em] p_X(1) \;&=\; \frac 1{6^3} \end{align} \]