Learning Notes

Stochastic – Common Distributions

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

GaussianDistribution

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.

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