F(\(x_i\)) = P(X ≤ \(x_i\))
Example
Consider an honest die with six faces that are identified by the Ace, King, Queen, Jack, and the numbers ten and two from a deck of cards. Consider a random variable defined by the function X from the set Ω of possible results on a set E of values attributed to these faces such as E = {100, 50, 20, 10, 5, 1}, as represented in this table:
| Result (A) | P(A) | X(A) |
| Ace | \(\frac{1}{6}\) | 100 |
| King | \(\frac{1}{6}\) | 50 |
| Queen | \(\frac{1}{6}\) | 20 |
| Jack | \(\frac{1}{6}\) | 10 |
| Ten | \(\frac{1}{6}\) | 5 |
| Two | \(\frac{1}{6}\) | 1 |
This means that a player earns 50 points if they draw a King, or one point if they draw a Two. The gains are distributed in the interval [1, 100].
The function F is the distribution function of the random variable X and F(\(x_i\)), where \(x_i\) ∈ [1, 100], is the probability of the event “the value of the random variable X is strictly less than \(x_i\)“.
The values of F are described in this table:
| \(x_i\) | F(\(x_i\)) = P(X < \(x_i\)) |
| \(x_i\) ≤ 1 | 0 |
| 1 < \(x_i\) ≤ 5 | \(\frac{1}{6}\) |
| 5 < \(x_i\) ≤ 10 | \(\frac{1}{6}+\frac{1}{6}=\frac{1}{3}\) |
| 10 < \(x_i\) ≤ 20 | \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{1}{2}\) |
| 20 < \(x_i\) ≤ 50 | \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{2}{3}\) |
| 50 < \(x_i\) ≤ 100 | \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{5}{6}\) |
| 100 < \(x_i\) | \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}= 1\) |