Cumulative density function

The phrase cumulative density function is most likely a confusion between

• cumulative distribution function and
• probability density function.

The value of a probability density function depends upon probabities of sets in arbitrarily small neighborhoods of that point (so it is not cumulative). The probability of finding a point between x₁ and x₂ is the integral of the probability density function over this range. (Probability mass functions, which assign positive probabilities to particular values, are sometimes referred to as discrete density functions.)

The cumulative distribution function is the integral (or, in the discrete case, the sum) for all values less than or equal to the current value of x, and so shows the accumulated probability so far. This is the sense in which it is cumulative.

For example, the probability density function of the normal distribution is a bell-curve, while the corresponding cumulative distribution function is a strictly increasing function that visually looks similar to a sigmoid function, which approaches 0 at −∞ and approaches 1 at +∞.