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4703-10-Notes-Mixtures

4703-10-Notes-Mixtures - Copyright c 2010 by Karl Sigman 1...

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Unformatted text preview: Copyright c 2010 by Karl Sigman 1 Simulating from mixtures Suppose that a cumulative distribution function (cdf) F ( x ) = P ( X ≤ x ) can be written as a mixture of two distinct cdfs; F ( x ) = pF 1 ( x ) + (1- p ) F 2 ( x ) , (1) where 0 < p < 1 is known, and both F 1 and F 2 are cdf’s from which we already have an algorithm for simulating from . (For example, F- 1 1 ( y ) and F- 1 2 ( y ) are explicitly known so that we can use the inverse transform method.) Then the following is an algorithm for simulating a rv X distributed as F : 1.1 Composition algorithm (C) for simulating from a two-point mixture: 1. Generate U . 2. If U ≤ p , then generate X distributed as F 1 ; otherwise ( U > p ) generate X distributed as F 2 Note that we can interpret p as the probability that F is distributed as F 1 , while 1- p is the probability that F is distributed as F 2 . So first we must decide (via a Bernoulli p trial) which of the two F 1 , F 2 to use, and then simulate from that. We say that the distribution of X is a mixture of the two distributions F 1 and F 2 . To see where such a mixture would naturally come from, imagine a box filled with two kinds (1,2) of new light bulbs, made by two different companies.with two kinds (1,2) of new light bulbs, made by two different companies....
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4703-10-Notes-Mixtures - Copyright c 2010 by Karl Sigman 1...

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