ELEC 321

# Three Important Distributions

Updated 2017-10-31

## Uniform

If random variable $X\sim\text{Uniform}(a, b)$ if and only if the PMF

$f(x)=\begin{cases} \frac{1}{b-a} & \text {if }a<x<b\\ 0&\text{otherwise} \end{cases}$

The distribution function is just the sum of the PMF from 0 to the value of interest.

The corresponding mean and variance is:

$\mu=\frac{b+a}{2}\\ \sigma^2=\frac{1}{12}(b-a)^2$

Example: application of the uniform distribution include the noise generated from a quantizer

## Exponential

Exponential random variables has the distribution function

$F(x)=1-e^{-\lambda x}$

The PMF is the derivative.

$f(x)=\lambda e^{-\lambda x}$

The distribution takes one parameter $\lambda$, which is the rate of occurrence. $\lambda >0$.

The associated mean and variance can be calculated as follows.

$\mu=\frac{1}{\lambda}\\ \sigma^2=\frac{1}{\lambda^2}$

The exponential random variables holds the the memoryless property: which states

$F(x+h)=F(h)$

## Gaussian / Normal

If random variable $X$ follows a normal distribution, $X\sim\text{N}(\mu, \sigma^2)$, its density function and distribution is as follows.

$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ $\Phi(x)=\frac{1}{2\pi}\int_{-\infty}^xe^{-\frac{s^2}{2}}\mathrm ds$