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

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 φ = tan−1(Z2/Z1).

1 2

(a) Find the joint density of A2 and φ, and from the result, conclude that A2 and φ are independent random variables, where A2 is a chi-

squared random variable with 2 df, and φ is uniformly distributed on (−π, π).

(b) Going in reverse from polar coordinates to rectangular coordinates, suppose we assume that A2 and φ are independent random variables, where A2 is chi-squared with 2 df, and φ is uniformly distributed

on (−π, π). With Z1 = A cos(φ) and Z2 = A sin(φ), where A is the

positive square root of A2, show that Z1 and Z2 are independent,

standard normal random variables.

Expert Solution

orchestra answered 1 year ago

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