63. The result of the previous exercise suggests how observed values of two independent standard normal variables can be generated by rst generating their polar coordinates with an exponential rv with and an independent uniform (0, 2p) rv: Let U1 and U2 be independent uniform (0, 1) rv s, and then let Show that the Zi s are independent standard normal.

Note: This is called the Box-Muller transformation after the two individuals who discovered it. Nowthat statistical software packages will generate almost instantaneously observations from a normal distribution with any mean and variance, it is thankfully no longer necessary for us to carry out the transformations just described let the software do it!