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	[latex]\frac{1}{6}[/latex]	100
King	[latex]\frac{1}{6}[/latex]	50
Queen	[latex]\frac{1}{6}[/latex]	20
Jack	[latex]\frac{1}{6}[/latex]	10
Ten	[latex]\frac{1}{6}[/latex]	5
Two	[latex]\frac{1}{6}[/latex]	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([latex]x_i[/latex]), where [latex]x_i[/latex] ∈ [1, 100], is the probability of the event "the value of the random variable X is strictly less than [latex]x_i[/latex]". The values of F are described in this table:

[latex]x_i[/latex]	F([latex]x_i[/latex]) = P(X < [latex]x_i[/latex])
[latex]x_i[/latex] ≤ 1	0
1 < [latex]x_i[/latex] ≤ 5	[latex]\frac{1}{6}[/latex]
5 < [latex]x_i[/latex] ≤ 10	[latex]\frac{1}{6}+\frac{1}{6}=\frac{1}{3}[/latex]
10 < [latex]x_i[/latex] ≤ 20	[latex]\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{1}{2}[/latex]
20 < [latex]x_i[/latex] ≤ 50	[latex]\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{2}{3}[/latex]
50 < [latex]x_i[/latex] ≤ 100	[latex]\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{5}{6}[/latex]
100 < [latex]x_i[/latex]	[latex]\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}= 1[/latex]