I need to generate random numbers following Normal distribution. If X is a radius, and Y is an angle, how I can transform it into Descartes coordinates, and are they normally distributed?
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What is the distribution of radii and angles? – Sycorax Dec 14 '15 at 20:19
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Seconding user777's comment. There is no way to know what the distribution of the rectangular coordinates will be if you do not specify the distribution of both the radius and angle. – Matthew Drury Dec 14 '15 at 20:55
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This is a classic method for generating 2 pseudorandom normal variates. – gung - Reinstate Monica Dec 14 '15 at 21:07
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In R2 angles is Unforim distributed on (0,2Pi), and for Radius I don't know. – user3568043 Dec 14 '15 at 20:30
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3This sounds like a request for the [Box-Muller method](http://stats.stackexchange.com/questions/132556). If not, please edit your post to clarify what you are looking for. – whuber Dec 14 '15 at 21:23
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Generate Y Uniform 0 to $2\pi$. Generate X from the exponential distribution.
Edit: you then may want to transform to Cartesian coordinates in the standard way. $$ A = X\sin(Y) \\ B = X\cos(Y) $$
Now A and B are distributed bivariate normal in the Cartesian coordinate system.
Ian Fellows
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2You're half way there. The OP wants normally distributed numbers. Right now you have 2 pseudorandom variates, neither of which are normal. Can you get the OP the rest of the way there? – gung - Reinstate Monica Dec 14 '15 at 21:04
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1Actually, (X,Y) is 1 bivariate random variable distributed normally and represented in the polar coordinate system. But I get your point and have amended the answer – Ian Fellows Dec 14 '15 at 23:55