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integration 143where dP is the probability of obtaining an x in the range x → x + dx. For theuniform distribution over [a, b], w(x)=1/(b − a).Inverse transform/change of variable method ⊙: Let us consider a change ofvariables that takes our original integral I (6.53) to the formI =∫ badx f(x)=∫ 10dWf[x(W )]w[x(W )] . (6.60)Our aim is to make this transformation such that there are equal contributionsfrom all parts of the range in W ; that is, we want to use a uniform sequenceof random numbers for W . To determine the new variable, we start with u(r),the uniform distribution over [0, 1],{1, for 0 ≤ r ≤ 1,u(r)=(6.61)0, otherwise.We want to find a mapping r ↔ x or probability function w(x) for whichprobability is conserved:w(x) dx = u(r) dr, ⇒ w(x)=dr∣dx∣ u(r). (6.62)This means that even though x and r are related by some (possibly) complicatedmapping, x is also random with the probability of x lying in x → x + dxequal to that of r lying in r → r + dr.To find the mapping between x and r (the tricky part), we change variablesto W (x) defined by the integral∫ xW (x)= dx ′ w(x ′ ). (6.63)−∞We recognize W (x) as the (incomplete) integral of the probability density u(r)up to some point x. It is another type of distribution function, the integratedprobability of finding a random number less than the value x. The functionW (x) is on that account called a cumulative distribution function and can also bethought of as the area to the left of r = x on the plot of u(r) versus r. It followsimmediately from the definition (6.63) that W (x) has the propertiesW (−∞)=0; W (∞)=1, (6.64)dW (x)dx= w(x), dW(x)=w(x) dx = u(r) dr. (6.65)Consequently, W i = {r i } is a uniform sequence of random numbers, and wejust need to invert (6.63) to obtain x values distributed with probability w(x).−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 143