Overview
The cumulative distribution function is a function that can be used to represent the likelihood of the outcomes of a random variable and used to calculate various statistics of that variable.
Definition
The cumulative distribution function is defined as
{% Prob(X \leq x) = F(x) %}
The function is required to have the following properties
- {% F(x) %} is right continuous, that is
{% \lim_{x \rightarrow x_0} F(x) = F(x_0) %}where {% x > x_0 %} in the limit.
-
{% \lim_{x \rightarrow - \infty} F(x) = 0 %}
-
{% \lim_{x \rightarrow \infty} F(x) = 1 %}
Example
The following chart demonstrates the typical features of a cumulative function, in particular, the monotonically increasing nature.