Learning Notes

# Introduction

Each random variables can have a different probability distribution. Such distribution has a very dominant effect on the behavior of a stochastic process as described in previous articles Stochastic – Poisson Process and Stochastic – Random Walk.

Some commonly used distribution are recorded here.

# Distributions

## Gaussian Distribution

$\displaystyle p(x) = \frac{1}{\sqrt{2\pi \sigma}} \exp \left[\frac{-(x - x_0)^2}{2\sigma^2} \right]$

$\displaystyle \langle p(x) \rangle = x_0 \indent \langle p(x)^2 \rangle = \sigma^2$

## Poisson Distribution

$\displaystyle p(n, t;\lambda) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}$

$\displaystyle \langle p(n, t;\lambda) \rangle = \lambda \indent \langle p(n, t;\lambda)^2 \rangle = \lambda$

## Bernoulli

$\displaystyle b(n) = \begin{cases} p, & n=1\\1-p,&i=0\\0,&\text{otherwise} \end{cases}$

$\displaystyle \langle b(n) \rangle = p \indent \langle b(x)^2 \rangle = p(1-p)$

## Geometric

$\displaystyle g(n) = (1-p)^{n-1} p, \forall n \geq 1$

$\displaystyle \langle g(n) \rangle = \frac{1}{p} \indent \langle g(x)^2 \rangle = \frac{1-p}{p^2}$

## Exponential

$\displaystyle p(x) = \lambda e ^{-\lambda x}, \forall x \geq 0$

$\displaystyle \langle p(x) \rangle = \frac{1}{\lambda} \indent \langle p(x)^2 \rangle = \frac{1}{\lambda ^2}$

## Gamma

$\displaystyle p(x;\alpha, n) = \frac{\alpha^n x^{n-1} e^{-\alpha x}}{\Gamma(n)}, \forall x \geq 0$

$\displaystyle \langle p(x;\alpha, n) \rangle = \frac{n}{\alpha} \indent \langle p(x;\alpha, n)^2 \rangle = \frac{n}{\alpha^2}$

## Rayleigh

$\displaystyle p(x;\sigma) = \frac{x}{\sigma^2} \exp \left[-\frac{x^2}{2 \sigma^2}\right], \forall r > 0$

$\displaystyle \langle p(x;\sigma) \rangle = \sigma \sqrt{\frac{\pi}{2}} \indent \langle p(x)^2 \rangle = \sigma^2 \left(2-\frac{pi}{2}\right)$

Properties: If two Gaussian variable X and Y are independent, then $\sqrt{X^2 +Y^2}$ has Rayleigh distribution of the same variance as X and Y.