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Properties

Every cumulative distribution function F X {\displaystyle F_{X}} is non-decreasing and right-continuous, which makes it a càdlàg function. Furthermore, lim x → − ∞ F X (x) = 0, lim x → + ∞ F X (x) = 1. {\displaystyle \lim _{x\to -\infty }F_{X}(x)=0,\quad \lim _{x\to +\infty }F_{X}(x)=1.}

Every function with these three 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 X {\displaystyle X} is a purely discrete random variable, then it attains values x 1, x 2, … {\displaystyle x_{1},x_{2},\ldots } with probability p i = p (x i) {\displaystyle p_{i}=p(x_{i})}, and the CDF of X {\displaystyle X} will be discontinuous at the points x i {\displaystyle x_{i}}: F X (x) = P (X ≤ x) = ∑ x i ≤ x P (X = x i) = ∑ x i ≤ x p (x i). {\displaystyle 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}).}

If the CDF F X {\displaystyle F_{X}} of a real valued random variable X {\displaystyle X} is continuous, then X {\displaystyle X} is a continuous random variable; if furthermore F X {\displaystyle F_{X}} is absolutely continuous, then there exists a Lebesgue-integrable function f X (x) {\displaystyle f_{X}(x)} such that F X (b) − F X (a) = P (a < X ≤ b) = ∫ a b f X (x) d x {\displaystyle F_{X}(b)-F_{X}(a)=\operatorname {P} (a<X\leq b)=\int _{a}^{b}f_{X}(x)\,dx} for all real numbers a {\displaystyle a} and b {\displaystyle b}. The function f X {\displaystyle f_{X}} is equal to the derivative of F X {\displaystyle F_{X}} almost everywhere, and it is called the probability density function of the distribution of X {\displaystyle X}.

If X {\displaystyle X} has finite L1-norm, that is, the expectation of | X | {\displaystyle |X|} is finite, then the expectation is given by the Riemann–Stieltjes integral E = ∫ − ∞ ∞ t d F X (t) {\displaystyle \mathbb {E} =\int _{-\infty }^{\infty }t\,dF_{X}(t)}

and for any x ≥ 0 {\displaystyle x\geq 0}, x (1 − F X (x)) ≤ ∫ x ∞ t d F X (t) {\displaystyle x(1-F_{X}(x))\leq \int _{x}^{\infty }t\,dF_{X}(t)} as well as x F X (− x) ≤ ∫ − ∞ − x (− t) d F X (t) {\displaystyle xF_{X}(-x)\leq \int _{-\infty }^{-x}(-t)\,dF_{X}(t)} as shown in the diagram with the two red rectangles. In particular, we have lim x → − ∞ x F (x) = 0, lim x → + ∞ x (1 − F (x)) = 0. {\displaystyle \lim _{x\to -\infty }xF(x)=0,\quad \lim _{x\to +\infty }x(1-F(x))=0.} In addition, the (finite) expected value of the real-valued random variable X {\displaystyle X} can be defined on the graph of its cumulative distribution function as illustrated by the drawing in the definition of expected value for arbitrary real-valued random variables.