Cumulative density function

Cumulative density function is a self-contradictory phrase resulting from confusion between:

• probability density function, and
• cumulative distribution function.

The two words cumulative and density contradict each other. The value of a density function in an interval about a point depends only on probabities of sets in arbitrarily small neighborhoods of that point, so it is not cumulative.

That is to say, if values are taken from a population of values described by the density function, and plotted as points on a linear axis, the density function reflects the density with which the plotted points will accumulate. The probability of finding a point between x₁ and x₂ is the integral of the probability density function over this range.

This is related to the probability mass function, which is the equivalent for variables that assign positive probability to individual points. The probability mass function is therefore sometimes referred to as the discrete density function.

In both cases, 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. Thus 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 +∞.