*X*will be strictly less than a given value.

F(\(x_i\)) = P(*X* ≤ \(x_i\))

*X*, and equal to one for all values \(x_i\) strictly greater than the greatest value of

*X*.

### 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 < \(x_i\) | \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}= 1\) |