Degrees of freedom and conditional expectations

Reference no: EM13108157

1. Show that if Z1 and Z2 are independent standard normal random variables, then for all ρ (correlation), Z₁ and ρZ₁+sqrt(1-ρ²)*Z₂are standard normal with correlation ρ.

2. 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)

Reference no: EM13108157

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