### Cumulative Distribution Function

#### Definition

The cumulative distribution function (cdf) gives the probability that the random variable $X$ is less than or equal to $x$ and is usually denoted $F(x)$. The cumulative distribution function of a random variable $X$ is the function given by $F(x)= \mathrm{P}[X \leq x].$

#### Proposition

Let $X$ be a random variable with cdf $F(x)$. Then $P[a<X \leq b] = F(b) - F(a).$

#### CDF of a Continuous Random Variable

Let $X$ be a continuous random variable with pdf $f(x)$. Then the cumulative distribution function $F(x)$ can be expressed by $F(x) =\int\limits_{- \infty}^x f(t) \mathrm{d} t.$

##### Properties

The cumulative distribution function $F(x)$ lies within $0$ and $1$ for all values of $x$. In addition to this $\lim\limits_{x \rightarrow -\infty}F(x) = 0$ and $\lim\limits_{x \rightarrow \infty}F(x) = 1.$

#### Video Example

In this video, Daniel Organisciak finds the cumulative distribution of a probability density function.

#### Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.