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.

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