Any help in this problem?


Suppose U and V are independent random variables with density f(u) and g(v) respectively.
The domain of U is the interval (0, 1) and the domain of V is v > 0. After the transformation


X = V sin(2U) 
Y = V cos(2U)


X and Y are independent, each following the standard normal distribution N(0, 1).


(a) Find f(u) and g(v).


(b) Show how to generate a normal random variable from uniform distribution without having to
do integration of normal density function.



Transformation problem involving 2 random variables
One obvious answer to (a) is:
g(v) is density of Rayleigh distribution and f(u) is density of uniform distribution on [0,1].


For (b), you need only noticed that V could be generated by 
V = \sqrt(-ln(W))
Here W is also uniform distributed on [0,1], the same as U. Nevertheless, W is independent of U.


This is well-known "Polar method" for generating pseudo-random number of normal variables. 


