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4703-10-Notes-ITM

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

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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] . If F is continuous, then F is invertible (since it is thus continuous and strictly increasing) in which case F - 1 ( y ) = min { x : F ( x ) = y } , the ordinary inverse function and thus F ( F - 1 ( y )) = y and F - 1 ( F ( x )) = x . In general it holds that F - 1 ( F ( x )) x and F ( F - 1 ( y )) y . F - 1 ( y ) is a non-decreasing (monotone) function in y . This simple fact yields a simple method for simulating a rv X distributed as F : Proposition 1.1 (The Inverse Transform Method) Let F ( x ) , x IR , denote any cumu- lative distribution function (cdf) (continuous or not). Let F - 1 ( y ) , y [0 , 1] denote the inverse function deﬁned in (1). Deﬁne X = F - 1 ( U ) , where U has the continuous uniform distribution over the interval

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

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