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The phrase '''''Cumulativecumulative density function''''' ismost alikely self-contradictoryis phrase resulting froma confusion between:
* ''[[probabilitycumulative densitydistribution function]],'' and
* ''[[cumulativeprobability distributiondensity 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 {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} is the integral of the probability density function over this range. ([[Probability mass function]]s, which assign positive probabilities to particular values, are sometimes referred to as ''discrete density functions''.)
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.
In both cases, theThe 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 {{math|''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 +∞. ▼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 {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} is the integral of the probability density function over this range.
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 +∞.
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 {{math|''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 +∞.
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