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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.

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  • $\begingroup$ Suppose you do your own homework problems? Voting to close. $\endgroup$
    – David Roberts ♦
    Commented Oct 30, 2012 at 5:15

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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.

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