Example 2.3.2.
A poll asks two yes/no questions. Let \(X\) indicate a yes on Question 1 and \(Y\) indicate a yes on Question 2. When the data is collected, the following joint distribution for \(X\) and \(Y\) is created:
| \(X = 0\) | \(X = 1\) | |
| \(Y = 0\) | 0.1 | 0.2 |
| \(Y = 1\) | 0.3 | 0.4 |
Note that the sum of all values in the table is 1; this is a probability distribution, and total probability must be 1. This table doesn’t show the distributions for \(X\) or \(Y\) individually. The data for the two distributions is mixed together. We can, if we wish, take this information and determine separate distributions for \(X\) and \(Y\text{.}\) For the distribution for \(X\text{,}\) we need to know \(\Pr(X = 0)\) and \(\Pr(X = 1)\) (with no reference to \(Y\)). We can get these probabilities by summing along the columns of the table:
\begin{align*}
\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \\
\Pr(X = 1) \amp = 0.2 + 0.4 = 0.6
\end{align*}
To get the distribution for \(Y\text{,}\) we should add along the rows:
\begin{align*}
\Pr(Y = 0) \amp = 0.1 + 0.2 = 0.3 \\
\Pr(Y = 1) \amp = 0.3 + 0.4 = 0.7
\end{align*}