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I am studying Probability theory and came to this exercise :

Let $U,V$ be independent uniform random variables over $[0,1]$. Show that $X:=\cos(2\pi V)\sqrt{-2\ln U}$ and $Y:=\sin(2\pi V)\sqrt{-2\ln U}$ are independent random variable which are normal distribution $N(0,1)$.

The proof is given in the textbook and uses the transfert theorem. And therefore, the probability density function of $X,Y$ are deducted from the Jacobian of the transformation.

My question is : Is this the usual method to solve such problems ? Could we use instead the cumulative distribution function or the characteristic function ? Or any other trick ?

Thanks for your help

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Each of these three methods is useful, the one to adopt in a given situation depends on (one's taste and on) the specifics of the problem. –  Did Jan 20 '13 at 9:10
    
Ok thanks, I think that's the kind of reflexes that are acquired with experience –  Alan Simonin Jan 20 '13 at 18:34
    
I wonder what "transfert" means.... –  Byron Schmuland Jan 21 '13 at 0:49
    
I didn't find a translation for that, it is called "Théorème du transfert" in french. $\mathbb E\left[\varphi(X)\right] \stackrel{\text{déf.}}{=} \int_\Omega \varphi \big(X(\omega)\big) \mathbb{P}(\mathrm d\omega) = \int_\mathbb{R} \varphi(x) \mathbb P_X(\mathrm dx)$ –  Alan Simonin Jan 21 '13 at 0:55
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