Marginal probabilities
Two sets of probabilities about the joint distribution of two discrete variables, \(X\) and \(Y\) are particularly important — marginal and conditional probabilities. (Conditional probabilities will be described on the next page.)
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 \(Y\).
\[ \begin{align} p_X(6) \;&=\; p(6,6) + p(6,5) + \cdots + p(6,1) \\ &=\; \frac 1{6^3} + \frac 1{6^2} + \frac 2{6^2} + \cdots + \frac 5{6^2} \\ &=\; \frac 1{6^3} + \frac {1+2+3+4+5}{6^2} \\ &=\; \frac 1{6^3} + \frac {15}{6^2} \\[0.7em] p_X(5) \;&=\; p(5,5) + p(5,4) + \cdots + p(5,1) \\ &=\; \frac 1{6^3} + \frac 1{6^2} + \frac 2{6^2} + \cdots + \frac 4{6^2} \\ &=\; \frac 1{6^3} + \frac {10}{6^2} \\[0.7em] p_X(4) \;&=\; \frac 1{6^3} + \frac {6}{6^2} \\[0.7em] p_X(3) \;&=\; \frac 1{6^3} + \frac {3}{6^2} \\[0.7em] p_X(2) \;&=\; \frac 1{6^3} + \frac {1}{6^2} \\[0.7em] p_X(1) \;&=\; \frac 1{6^3} \end{align} \]This is illustrated in the following 3-dimensional bar chart of the joint probabilities.
Click Marginal for X to stack the bars and find the marginal probabilities for \(X\). Click the middle rotation button to rotate the 3-dimensional diagram to display a conventional bar chart for the marginal distribution of \(X\).
Drag the centre of the diagram to rotate back, then click Joint probs, followed by Marginal for Y. This shows the corresponding marginal distribution of the minimum value from the three dice, \(Y\).