An important property of a cumulative distribution function F x (x) of a random variable is:

The CDF F x in every case is non-decreasing and right-continuous

lim_{x→-∞}F_{x}(x) = 0 and lim_{x→+∞}F_{x}(x) = 1

Suppose A and B are real numbers, and X is a continuous random variable. Thus, F x is equal to the derivative of F x, such that

Assuming that X is a discrete random variable, then it will take the values x 1 , x 2 , x 3 ,… with probability p i = p(x i ), and the CDF of X will be discontinuous at the points x i:

FX(x) = P(X ≤ x) = ∑xi≤xP(X=xi)=∑xi≤xp(xi)∑xi≤xP(X=xi)=∑xi≤xp(xi)

Real values are defined by this function, sometimes implicitly rather than explicitly. A CDF is a fundamental concept of PDF (Probability Distribution Function)

CDF can be illustrated by rolling a fair six-sided die, where X is the random variable.

When a six-sided die is rolled, the probability of getting an outcome is given as follows:

Number of chances obtaining1 = P(X≤ 1 ) = 1 / 6

Number of chances obtaining 2 = P(X≤ 2 ) = 2 / 6

Number of chances obtaining 3 = P(X≤ 3 ) = 3 / 6

Number of chances obtaining 4 = P(X≤ 4 ) = 4 / 6

Number of chances obtaining 5 = P(X≤ 5 ) = 5 / 6

Number of chances obtaining 6 = P(X≤ 6 ) = 6 / 6 = 1