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[this entry is currently being revised, so hold off on corrections until this line is removed]
Let
 . Then $F$ is a distribution function if
- $F$ is nondecreasing,
- $F$ is continuous from the right,
- $\lim_{x \rightarrow -\infty} F(x) = 0$ , and $\lim_{x \rightarrow \infty} F(x) = 1$ .
As an example, suppose that
 and that $\mathcal{B}$ is the $\sigma$ -algebra of Borel subsets of  . Let $P$ be a probability measure on $(\Omega, \mathcal{B})$ . Define $F$ by $$ F(x) = P((-\infty, x]). $$ This particular $F$ is called the distribution function of $P$ . It is easy to verify that 1,2, and 3 hold for this $F$ .
In fact, every distribution function is the distribution function of some probability measure on the Borel subsets of  . To see this, suppose that $F$ is a distribution function. We can define $P$ on a single half-open interval by $$ P((a,b]) = F(b) - F(a) $$ and extend $P$ to unions of disjoint intervals by $$ P( \cup_{i=1}^\infty (a_i, b_i])= \sum_{i=1}^\infty P((a_i, b_i]).
$$ and then further extend $P$ to all the Borel subsets of  . It is clear that the distribution function of $P$ is $F$ .
Suppose that $(\Omega, \mathcal{B}, P)$ is a probability space and
 is a random variable. Then there is an induced probability measure $P_X$ on  defined as follows:
$$ P_X(E) = P(X^{-1}(E)) $$ for every Borel subset $E$ of  . $P_X$ is called the distribution of $X$ . The distribution function of $X$ is $$ F_X(x) = P(\omega | X(\omega) \leq x). $$ The distribution function of $X$ is also known as the law of $X$ . Claim: $F_X$ = the distribution function of $P_X$ .
\begin{eqnarray*} F_X(x) &=& P(\omega | X(\omega) \leq x) \\ &=& P(X^{-1}((-\infty, x]) \\ &=& P_X((-\infty, x]) \\ &=& F(x). \end{eqnarray*}
Suppose that
 is a nonnegative function such that $$ \int_{-\infty}^\infty f(t)dt=1. $$ Then one can define
 by $$ F(x) = \int_{-\infty}^x f(t)dt. $$ Then it is clear that $F$ satisfies the conditions 1,2,and 3 so $F$ is a distribution function. The function $f$ is called a density function for the distribution $F$ .
If $X$ is a discrete random variable with density function $f$ and distribution function $F$ then
$$F(x)=\sum_{x_j\leq x} f(x_j).$$
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