Cumulative distribution function

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A Cumulative Distribution Function
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In probability theory, a cumulative distribution function F(x) of a probability density function say f(x) is a real valued and continuous function whose value is the proportion of probability values of a variable which occur on the part of the real line up and including the value of that variable; i.e.,



Considering this definition in light of the Fundamental Theorem of Calculus yields:



Due to the properties of the probability density function f(x), the cumulative distribution function F(x) will have the following properties:


(1) inside its domain of support.


(2) , i.e., finitely convergent (to unity by convention).


(3) for a<b, i.e., is non-decreasing


If the domain of the variable is finite, then the upper limit in equation (2) above should be the upper bound of the variables domain of support.