This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 cdfs 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 twopoint 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....
View
Full
Document
This note was uploaded on 10/17/2010 for the course IEOR 4703 taught by Professor Sigman during the Fall '07 term at Columbia.
 Fall '07
 sigman

Click to edit the document details