Difference between revisions of "Random Variable Distributions"

Revision as of 00:30, 25 October 2013

This page describes random distributions supported by Filebench

Uniform Distribution

DOMAIN: [a;b]

<math> f(x; a, b) = {1 \over {b - a}} </math>

<math> F(x; a, b) = {x - a \over {b - a}} </math>

<math> E(a,b) = {(b - a) \over 2} </math>

Exponential Distribution

DOMAIN: <math> [0:+\infty) </math>

<math> f(x; \lambda) = \lambda e^{-\lambda x} </math>

<math> F(x; \lambda) = 1 - e^{-\lambda x} </math>

<math> E(\lambda) = {1 \over \lambda} </math>

Erlang and Gamma Distributions

DOMAIN: <math> [0:+\infty) </math>

<math> f(x; k, \lambda) = {\lambda^{k} x^{k-1} e^{-\lambda x} \over \Gamma(k)} </math>

<math> F(x; k, \lambda) = {\gamma(k, \lambda x) \over \Gamma(k)} </math>

<math> E(k,\lambda) = {k \over \lambda} </math>

In Erlang distribution <math>k</math> is an integer. In Gamma distribution <math>k</math> is a real number.

NOTICE: <math>\gamma()</math> above is a non-normalized incomplete gamma function. Gnuplot's <math>igamma()</math> function, however, is already normalized (i.e., divided by <math>\Gamma(k)</math>).

Weibull Distribution

DOMAIN: <math> [0:+\infty] </math>

<math> f(x; k, \lambda) = {k \over \lambda} \left({x \over \lambda}\right)^{k - 1} e^{\left({- {x \over \lambda}}\right) ^ k} </math>

<math> f(x; k, \lambda) = 1 - e^{\left(-{x \over \lambda}\right)^k}</math>

<math> E(k, \lambda) = \lambda \Gamma(1 + {1 \over k})</math>

Normal Distribution

DOMAIN: <math> [-\infty:+\infty] </math>

<math> f(x; \mu, \sigma) = {1 \over \sigma \sqrt{2\pi}} e^{- {(x - \mu)^2 \over 2\sigma^2}} </math>

<math> f(x; \mu, \sigma) = {1 \over 2} \left[ 1 + erf\left({ x - \mu \over \sqrt{2 \sigma^2}}\right) \right] </math>

<math> E(\mu, \sigma) = \mu </math>

Log-normal Distribution

PDF: <math> f(x;\mu,\sigma) = {1 \over x \sqrt{2\pi\sigma}} e^{(ln x - \mu)^2 \over 2 \sigma^2} </math>

CDF: <math> F(x;\mu,\sigma) = {1 \over 2} + {1 \over 2} erf \left[ {ln x - \mu \over \sigma \sqrt{2}} \right]</math>

EXPECTED: <math> E(\mu, \sigma) = e^{\mu + {\sigma^2 \over 2}} </math>