With reference to equations (4.2)and (4.3), let Z1 = U1 and Z2 =?U2 be independent, standard normal variables.Consider the polar coordinates of the point (Z1,Z2), that is,
A2 = Z2 + Z2
and ? =tan?1(Z2/Z1).
1 2
(a) Find the joint density of A2 and?, and from the result, conclude that A2 and? are independent random variables, where A2 is achi-
squared random variable with 2 df, and ? is uniformlydistributed on (??, ?).
(b) Going in reverse from polar coordinates torectangular coordinates, suppose we assume that A2 and? are independent random variables, where A2 ischi-squared with 2 df, and ? is uniformly distributed
on (??, ?). With Z1 =A cos(?) and Z2 = Asin(?), where A is the
positive square root of A2, show that Z1 andZ2 are independent,
standard normal random variables.
