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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 |
There is no such thing as a "discrete value". Any number at all can be a value of a discrete probability distribution. Discrete probability distributions are those that assign positive probabilities to individual numbers. |
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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 [[
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.
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