The cumulative distribution function of a random variable is a function that, when evaluated at a point , gives the probability that the random variable will take on a value less than or equal to . For example, a random variable representing a single dice roll has cumulative distribution function
For a discrete random variable, the cumulative distribution is defined by
Furthermore, the p.d.f. is related to the cumulative distribution by
where is the next smallest possible value of . In the case of a random variable defined on integers (as is typical), . This forms the intuition for the relationship between the continuous p.d.f. and continuous cumulative distribution, where the p.d.f. is the derivative of the c.d.f.