I have a problem with this explanation, the book says we can generate Y1 Y2 from X1 X2 but when we want to answer questions about the distribution of X1 X2 we take arctan which returns values in range (-PI/2,PI/2) even though we X1 X2 range is (0,1) I understand it makes no difference in the part when we calculate uniform pdf from inverse since it is 1 for each X1 X2 but I can't understand how it works in Jacobian. I think author made jump here and didn't want to go into details
2026-03-26 02:53:02.1774493582
Generating Two Independent Standard Normal Variables
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Actually the inverse trigonometric functions can be defined with several ranges. In your formula, it's clear that they are taking the range $[0,2\pi]$
Regarding the Jacobian (and the proof in general) of this Box-Muller transform, there are many answers:
Box-Muller Transform Normality
proof of Box–Muller transform (polar form)
https://stats.stackexchange.com/questions/188183/some-details-about-the-box-muller-transform-method