Suppose that X can take any one of n values \(x_{1}, x_{2},\cdots ,x_{n}\) and Y can take any one of m values \(y_{1}, y_{2}, \cdots, y_{m}\).
Then the probability of the event that \(X=x_{i}\) and \(Y=y_{j}\) is given by：
$$f(x_{i}, y_{j})=P(X=x_{i}, Y=y_{j})$$
and \(f(x, y)\) is called a joint probability function for X and Y.
The probabilities of \(X=x_{i}\) or \(Y=y_{j}\) are given by
\begin{eqnarray*}f_{X}(x_{i})&=&P(X= x_{i})=\sum_{j=1}^{m} f(x_{i}, y_{j})\\f_{Y}(y_{j})&=&P(Y= y_{i})=\sum_{i=1}^{n} f(x_{i}, y_{j})\end{eqnarray*}
and they are called the marginal probability functions of X and Y, respectively.
Note that \(\sum_{i=1}^{n} f_{X}f(x_{i})=1\)
$$\sum_{i=1}^{n} f_{X}(x_{i})=1\ \ \ \ \ \sum_{j=1}^{m} f_{Y}(y_{j})=1$$
which can be written as \(\sum_{i=1}^{n}\sum_{j=1}^{m} f(x_{i}, y_{j})=1\).
Example
The Table represents the joint probability functions for X and Y.
Find the marginal probability functions of X and Y.
| X=1 | X=2 | |
| Y=-1 | \(\frac{2}{3}\) | \(\frac{1}{6}\) |
| Y=1 | \(\frac{1}{6}\) | \(0\) |
Solution : The marginal probability functions of X and Y are given as below:
\(P(X= 1)=\displaystyle\sum_{y} f(1, y)=P(X=1, Y=-1)+P(X=1, Y=1)=\frac{2}{3}+\frac{1}{6}=\frac{5}{6}\) \(P(X= 2)=\displaystyle\sum_{y} f(2, y)=P(X=2, Y=-1)+P(X=2, Y=1)=0+\frac{1}{6}=\frac{1}{6}\) \(P(Y= -1)=\displaystyle\sum_{x} f(x, -1)=P(X=1, Y=-1)+P(X=2, Y=-1)=\frac{2}{3}+0=\frac{2}{3}\) \(P(Y= 1)=\displaystyle\sum_{x} f(x, 1)=P(X=1, Y=1)+P(X=2, Y=1)=\frac{1}{6}+\frac{1}{6}=\frac{1}{3}\)| X=1 | X=2 | Marginal Probability P(Y=y) | |
| Y=-1 | \(\frac{2}{3}\) | \(0\) | \(\frac{2}{3}\) |
| Y=1 | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{3}\) |
| Marginal Probability P(X=x) | \(\frac{5}{6}\) | \(\frac{1}{6}\) | \(1\) |
Discrete Case
Let X and Y be discrete random variables. Then the marginal probability functions are given by
$$\begin{eqnarray*}f_{X}(x)&=&P(X= x)=\sum_{y} f(x, y)\\f_{Y}(y)&=&P(Y= y)=\sum_{x} f(x, y)\end{eqnarray*}$$
where \(f(x,y)\) is the joint probability function of X and Y.
Continuous Case
Let X and Y be continuous random variables. Then the marginal probability functions are given by
\begin{eqnarray*}f_{X}(x)&=&P(X= x)=\int_{-\infty}^{\infty} f(x, y)\ dy\\f_{Y}(y)&=&P(Y= y)=\int_{-\infty}^{\infty} f(x, y)\ dx\end{eqnarray*}
where \(f(x,y)\) is the joint probability function of X and Y.