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Every function with these four properties is a CDF, i.e., for every such function, a [[random variable]] can be defined such that the function is the cumulative distribution function of that random variable.
If <math>X</math> is a purely [[discrete random variable]], then it attains values <math>x_1,x_2,\ldots</math> with probability <math>p_i =
:<math>F_X(x) = \operatorname{P}(X\leq x) = \sum_{x_i \leq x} \operatorname{P}(X = x_i) = \sum_{x_i \leq x} p(x_i).</math>
for all real numbers <math>a</math> and <math>b</math>. The function <math>f_X</math> is equal to the [[derivative]] of <math>F_X</math> [[almost everywhere]], and it is called the [[probability density function]] of the distribution of <math>X</math>.
== Examples ==
As an example, suppose <math>X</math> is [[Uniform distribution (continuous)uniformly distributed]] on the unit interval <math>[0,1]</math>.
