### 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 \[\lim\limits_{x \rightarrow \infty}F(x) = 1.\]

Observe, for $a**fundamental theorem of calculus, we see that $F$ is an anti-derivative of $f$, i.e. \[\dfrac{\mathrm{d} \;}{\mathrm{d} x} F(x) = f(x).\] hence**

- $F$ is continuous,
- $F$ is non-decreasing (because its derivative $f$ is non-negative).

#### Worked Example

###### Worked Example

The pdf of a continuous random variable $X$ is \[f(x) = \begin{cases} 0 & \text{ if } x<1, \\[6 pt] \dfrac{2}{x^3} &\text{ otherwise}. \end{cases}\]

Find the cumulative distribution function $F(x)$.

###### Solution

We need to split this up into two parts, where $x<1$ and where $x \geq 1$. Substitute into the formula provided in the definition for each part.

If $x<1$ we have \[\int\limits_{-\infty}^{x}f(t) \mathrm{d} t =0\]

and if $x\geq 1$ we have \begin{align} \displaystyle\int\limits_{-\infty}^{x}f(t)\mathrm{d} t &= \int\limits_{1}^{x}f(t) \mathrm{d} t \\ & = \int\limits_{1}^{x} \frac{2}{t^3}\mathrm{d} t \\ & = \left[ - \frac{1}{t^2} \right]_{1}^{x} \\ & = -\frac{1}{x^2} + 1\text{.} \end{align}

So we have obtained \[F(x) = \begin{cases} 0 & \text{ if } x<1, \\[6pt] 1-\frac{1}{x^2} &\text{otherwise} . \end{cases}\]

Note that $\lim\limits_{x \rightarrow -\infty} F(x) = 0$ and $\lim\limits_{x \rightarrow \infty} F(x) =1$ as required.

#### 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.

#### External Resources