distributed (as would be expected from the radial symmetryand the radius has pdf fr (rr - / moreoverr and th are independent we will see shortlyin ex ample that has the same distribution as - ln with ( soto simulate xy ~iid ( )the idea is to first simulate and th independently and then return cos(thand sin(thas pair of independent standard normal random variables this leads to the box-muller approach for generating standard normal random variables algorithm normal random variable simulationbox-muller approach outputindependent standard normal random variables and simulate two independent random variablesu and from ( / (- ln cos( pu / (- ln sin( pu return xy cholesky decomposition once standard normal number generator is availablesimulation from any ndimensional normal distribution (usis relatively straightforward the first step is to find an matrix that decomposes into the matrix product bbin fact there exist many such decompositions one of the more important ones is the cholesky decompositionwhich is special case of the lu decompositionsee section for more information on such decompositions in pythonthe function cholesky of numpy linalg can be used to produce such matrix once the cholesky factorization is determinedit is easy to simulate (usasby definitionit is the affine transformation bz of an -dimensional standard normal random vector algorithm normal random vector simulation inputus outputx (us determine the cholesky factorization bb simulate [ zn by drawing zn ~iid ( bz return example (simulating from bivariate normal distributionthe python code below draws iid samples from the two bivariate ( normal pdfs in figure the resulting point clouds are given in figure bvnormal py import numpy as np from numpy random import randn import matplotlib pyplot as plt change to for other plot sigma np array ([[ ][ ]]