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trying to explain the origins of the terms cumulative and density in order to make sure readers find the correct article; introducing the discrete version as its article is otherwise hard to find |
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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 {{math|x<sub>1</sub>}} and {{math|x<sub>2</sub>}} 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 can only take discrete values. 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 sigmoid rising from {{math|P ≈ 0}} at the extreme left, to {{math|P ≈ 1}} at the extreme right.
{{disambig}}
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