# Random Variables

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)

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

#### Solution Preview

Please see attachment.

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)