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1. (a) Show that if Z1 and Z2 are independent standard normal random variables, then for all ρ (correlation), Z1 and ρZ1+sqrt(1-ρ2)*Z2 are standard normal with correlation ρ.

(b) Show that for all ρ and v, T1 = (Z1)/sqrt(X/v)and T2 = (Z2)/sqrt(X/v)
have correlation ρ, where Z1and Z2 are standard normal with correlation ρ, and X is independent of both Z1 and Z2 and has a Chi-Squared distribution. (for simplification use the fact that T1 and T2 both have the t distribution with v degrees of freedom and conditional expectations.)
(v=degrees of freedom)

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Solution Summary

Correlation proofs are provided. The solution is detailed and well presented.