Convolution random number generator: Difference between revisions

Revision as of 10:21, 30 January 2009

Revision as of 12:19, 20 October 2009

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== Example ==

Consider the random variable <math>X\ \sim \operatorname{Erlang}(k, \theta)</math>, defined as the sum of ''k'' random variables each with an [[exponential distribution]] <math>\operatorname{Exp}(k \theta) \,</math>:see [[Gamma distribution]].

Notice that:

:<math>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 \exp(k \theta)</math> then <math>X=\sum_{i=1}^k X_i \sim \operatorname{Erlang}(k,\theta) .</math>