Box Muller Transformation

When I read the post, I got an intuitive understanding of Box Muller Transformation because it has a lot of images.

However I didn’t think I fully understand Box-Muller transformation and I didn’t understand the reason why we need to draw two samples for the transformation.

I read the proof written by Box and Muller. It’s difficult for me to understand it due to lack of statistics knowledge. For example,

If Uh as a rectangular deinsity(uniform distribution) on (0, 10) then has a Chi-squared distribution with two degrees of freedom.

Probably it’s trivial for statistician. But I don’t know Chi-squared distribution…

Then I googled about Box Muller transformation again and found a proof which is easier to understand.

The proof uses basic math formula transformation and explain the important trick in the first paragraph, which gives us the reason why we need to draw two samples, in other words, why we need to consider joint probability of two normal distribution.