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1. Show that if Z1 and Z2 are independent standard normal random variables, then for all ρ (correlation), Z_{1} and ρZ_{1}+sqrt(1-ρ^{2})*Z_{2 }are standard normal with correlation ρ.
2. Show that for all ρ and v, T_{1} = (Z_{1})/sqrt(X/v)and T_{2} = (Z_{2})/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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