correlated random variables

gradnu

I have two independent standard normal random variables X1,X2. Now I want to construct two new normal random variables Y1,Y2 with mean[tex]\mu[/tex]1, [tex]\mu[/tex]2 and variance ([tex]\sigma[/tex]1)^2, ([tex]\sigma[/tex]2)^2 and correlation [tex]\rho[/tex].
How do I approach this problem?

mathman

Y₁=s₁X₁+m₁
Y₂=bX₁+cX₂+m₂
where b²+c²=s₂²
b=rs₂, therefore c=s₂(1-r²)^1/2

gradnu

Thanks mathman.
But what was your thought process? How did you come up with these relations?

mathman

From long past experience I know that to get correlated normal variables from uncorrrelated standard normal, you just need a linear combination. Adding the desired means is obvious. Also since there are four free coefficients and there are only three conditions, I just set one coefficient to 0.