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160b_ch2

160b_ch2 - Spring 2010 Pstat160b H-O#5 Simulation of...

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Spring 2010 Pstat160b H-O #5 Simulation of Continuous Random Variables The starting point of all simulations methods is uniform U (0 , 1) random variable. Those are obtained from (pseudo) random number generator, which we assume are available to us. Note that any uniform random variables U ( a,b ) can be generated from those by setting X = a + ( b - a ) * U . Also Bernoulli random variables (value 1 with probability p , 0 with probability 1 - p ) can be generated by setting X = 1 if U < p , and X = 0 if U > p . Inverse transform method Suppose we want to simulate a continuous random variable with C.d.f F X ( x ) (here to simplify notation we just write F ( x ). Since a C.d.f is increasing and here assumed to be continuous, its inverse function F - 1 is well deﬁned as (see also graphs below). If F is strictly increasing, there is no ambiguity and F - 1 is given by F - 1 ( y ) = x such that F ( x ) = y If there are intervals on which F is “ﬂat”, then all the x in that interval return the same y . To be well deﬁned, we chose the smallest such x as the value of the inverse. The inverse transform algorithm just consists of selecting U , a uniform U (0 , 1) random variable, and set X = F - 1 ( U ). Note that if F is constant on some interval, this deﬁnition give a correct answer, regardless of the particular choice of the inverse since the probability of a single point for the uniform U (0 , 1) is 0 anyway. -

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160b_ch2 - Spring 2010 Pstat160b H-O#5 Simulation of...

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