Convolution random number generator: Difference between revisions

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Revision as of 10:11, 10 November 2010

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In [[statistics]] and [[computer software]], a '''convolution random number generator''' is a type of [[random number generator]] that can be used to generate [[random variatesvariate]]s from certain classes of [[probability distribution]]. The particular advantage of this type of approach is that it allows advantage to be taken of existing software for generating random variates from other, usually non-uniform, distributions. However, faster algorithms may be obtainable for the same distributions by other more complicated approaches.

A number of distributions can be expressed in terms of the (possibly weighted) sum of two or more [[random variablesvariable]]s from other distributions. (The distribution of the sum is the [[convolution]] of the distributions of the individual random variables).

== Example ==

Consider the problem of generating a random number correspobding to a random variable with an [[Erlang distribution]], <math>X\ \sim \operatorname{Erlang}(k, \theta)</math>,. Such a random variable can be defined as the sum of ''k'' random variables each with an [[exponential distribution]] <math>\operatorname{Exp}(k \theta) \,</math>:see. This problem is equivalent to generating a random number for a special case of the [[Gamma distribution]], in which the [[shape parameter]] takes an integer value.

Notice that:

:<math>\operatorname{E}[X] = \frac{1}{k \theta} + \frac{1}{k \theta} + \cdots + \frac{1}{k \theta} = \frac{1}{\theta} .</math>

One can now generate <math>\operatorname{Erlang}(k, \theta)</math> samples using a random number generator for the exponential distribution:

if <math>X_i\ \sim \expoperatorname{Exp}(k \theta)</math>&nbsp; &nbsp; then <math>X=\sum_{i=1}^k X_i \sim \operatorname{Erlang}(k,\theta) .</math>

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[[Category:Non-uniform random numbers]]