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    Random Variables

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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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    https://brainmass.com/statistics/normal-distribution/random-variables-for-independent-samples-16476

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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.
    Proof. Since Z1 and Z2 are independent standard normal random variables, we have
    E(Z1)=E(Z2)=0, Var(Z1)=Var(Z2)=1 , E(Z1Z2)=E(Z1)E(Z2)=0.
    We know that . Similarly, . So,

    We now try to find

    So, the correlation coefficient of Z1 and is

    (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 ...

    Solution Summary

    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)

    $2.49

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