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144 chapter 6The crux of this technique is being able to invert (6.63) to obtain x = W −1 (r).Let us look at some analytic examples to get a feel for these steps (numericalinversion is possible and frequent in realistic cases).Uniform weight function w: We start with the familiar uniform distributionw(x)={ 1b−a, if a ≤ x ≤ b,0, otherwise.(6.66)After following the rules, this leads toW (x)=∫ xadx ′ 1b − a = x − ab − a(6.67)⇒ x = a +(b − a)W ⇒ W −1 (r)=a +(b − a)r, (6.68)where W (x) is always taken as uniform. In this way we generate uniformrandom 0 ≤ r ≤ 1 and uniform random a ≤ x ≤ b.Exponential weight: We want random points with an exponential distribution:{ 1λw(x)=e−x/λ , for x>0,0, for x0. Notice thatour prescription (6.53) and (6.54) tells us to use w(x)=e −x/λ /λ to remove theexponential-like behavior from an integrand and place it in the weights andscaled points (0 ≤ x i ≤∞). Because the resulting integrand will vary less, itmay be approximated better as a polynomial:∫ ∞0dx e −x/λ f(x) ≃ λ NN∑f(x i ), x i = −λ ln(1 − r i ). (6.70)i=1Gaussian (normal) distribution: We want to generate points with a normaldistribution:w(x ′ )= 1 √2πσe −(x′ −x) 2 /2σ 2 . (6.71)This by itself is rather hard but is made easier by generating uniform distributionsin angles and then using trigonometric relations to convert themto a Gaussian distribution. But before doing that, we keep things simple by−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 144