How to Construct Correlated Normal Variables from Independent Normals?

[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$$\mu$$1, $$\mu$$2 and variance ($$\sigma$$1)^2, ($$\sigma$$2)^2 and correlation $$\rho$$.
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]

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

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.