Transcription of 1 Inverse Transform Method - Columbia University
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1 Inverse Transform Method - Columbia University
Copyright c 2010 by Karl Sigman 1 Inverse Transform Method Assuming our computer can hand us, upon demand, iid copies of rvs that are uniformly dis- tributed on (0, 1), it is imperative that we be able to use these uniforms to generate rvs of any desired distribution (exponential, Bernoulli etc.). The first general Method that we present is called the Inverse Transform Method . Let F (x), x IR, denote any cumulative distribution function (cdf) (continuous or not). Recall that F : IR [0, 1] is thus a non-negative and non-decreasing (monotone) function that is continuous from the right and has left hand limits, with values in [0, 1]; moreover F ( ) = 1. and F ( ) = 0. Our objective is to generate (simulate) rvs X distributed as F ; that is, we want to simulate a rv X such that P (X x) = F (x), x IR. Define the generalized Inverse of F , F 1 : [0, 1] IR, via (1) F 1 (y) = min{x : F (x) y}, y [0, 1].
is the counting process of a Poisson process at rate , then N(1) has a Poisson distri-bution with mean . Thus if we can simulate N(1), then we can set X= N(1) and we are done. Let Y = N(1) + 1, and let t n = X 1 + + X n denote the nth point of the Poisson process; the X i are iid with an exponential distribution at rate . Note that Y = minfn 1 : t
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