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integration 145realizing that we can obtain (6.71) with mean x and standard deviation σ byscaling and a translation of a simpler w(x):w(x)= 1 √2πe −x2 /2 , x ′ = σx + x. (6.72)We start by generalizing the statement of probability conservation for twodifferent distributions (6.62) to two dimensions [Pres 94]:p(x, y) dx dy = u(r 1 ,r 2 ) dr 1 dr 2 ⇒ p(x, y)=u(r 1 ,r 2 )∂(r 1 ,r 2 )∣ ∂(x, y) ∣ .We recognize the term in vertical bars as the Jacobian determinant:J =∂(r 1 ,r 2 )∣∣ ∂(x, y)= ∂r 1∂x∣ def∂r 2∂y − ∂r 2 ∂r 1∂x ∂y . (6.73)To specialize to a Gaussian distribution, we consider 2πr as angles obtainedfrom a uniform random distribution r, and x and y as Cartesian coordinatesthat will have a Gaussian distribution. The two are related byx = √ −2lnr 1 cos 2πr 2 , y = √ −2lnr 1 sin 2πr 2 . (6.74)The inversion of this mapping produces the Gaussian distributionr 1 = e −(x2 +y 2 )/2 , r 2 = 1 y +y 2 )/22π tan−1 x , J = . (6.75)−e−(x22πThe solution to our problem is at hand. We use (6.74) with r 1 and r 2 uniformrandom distributions, and x and y are then Gaussian random distributionscentered around x =0.−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 145