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  3.       with reference to equations (4.2) and (

      With reference to equations (4.2) and (4.3), let Z1 = U1 and Z2 = ?U2 be...

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Statistics

      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.

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