# Cumulative Distribution Function Motivation: A general concept in [[Estimation Theory]] A distribution function of a real variable $x$, taking at each $x$ the value equal to the probability of the inequality $X < x$. ## Properties of a Distribution Function 1. $F(x') \leq F(x'')$ when $x' < x''$ 2. $F(x)$ is continuous for all $x$ 3. $\lim_{x \to -\infty} F(x) = 0$, $\lim_{x \to \infty} F(x) = 1$