Note 3 - Y has any distribution G then X = G Y will have a...

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Math 151 – Probability Spring 2011 Jo Hardin Functions of random variables: simulation The motivation We can simulate real numbers on the interval [0,1]. We’d like to be able to simulate variables from other distributions. In fact, we’d like to be able to simulate observations from the following distribution: pdf: g ( ? ) = λe - ? 0 cdf: G ( ? ) = 1 - e - ? 0 The set up Let X be a uniform [0,1] random variable. That is, f X ( x ) = 1 0 x 1; F X ( x ) = x 0 x 1. Let Y = G - 1 ( X ). What is the distribution of Y? Note: X = 1 - e - Y λ Y = - ln(1 - X ) The solution F Y ( y ) = P ( Y y ) = P ( G - 1 ( X ) y ) = P ( X G ( y )) = F X ( G ( y )) = G ( y ) That is, if we let Y = G - 1 ( X ), then the random variable Y will have exactly the distribution for which we were hoping. The implications The relationship above holds in both directions. That is, if
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Unformatted text preview: Y has any distribution G , then X = G ( Y ) will have a uniform distribution on [0,1]. F X ( x ) = P ( X x ) = P ( G ( Y ) x ) = P ( Y G-1 ( x )) = G ( G-1 ( x )) = x x 1 Which proves that X has a uniform distribution on [0,1]. How does it work? 1. (a) Find a random uniform observation, x * (b) G-1 ( x * ) will be the random exponential observation we simulate. 2. (a) Find a random observation from any distribution, y * (b) G ( y * ) will be the random uniform [0,1] observation we simulate...
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This note was uploaded on 02/17/2012 for the course MATH 151 taught by Professor Jo.h during the Spring '10 term at Pomona College.

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