Generating independent random variables from correlated random variables

Question

I have 2 standard normal, bivariate correlated random variables, $corr \ (X_1, X_2)=\rho$.

I want to generate two independent standard normal random variables from these 2.

I tried to use what I learned from my previous thread: How does the formula for generating correlated random variables work?

I got the following:

$0=cov(\alpha X_1+\beta X_2, X_1)=\alpha\cdot cov(X_1X_2)+\beta\cdot\rho=\alpha+\beta\rho$

and

$\alpha^2+\beta^2=1$

But then I can't proceed. Can someone help me with that?

Answer 1

You say you need solve $$0=\alpha+\beta\rho\ (1)$$ $$\alpha^2 + \beta^2=1\ (2)$$

However, I think (2) is improper. You want to get two independent standard normal random variables, which are $X_1$ and $Y=\alpha X_1 + \beta X_2$. $X_1$ is made to follow standard normal. How about $Y$? $Y$ follows normal distribution, the mean of $Y$ is apparently 0, and the variance of $Y$ should be 1, so you need $1=var(Y)=\alpha^2 var(X_1)+\beta^2 var(X_2)+2\alpha\beta cov(X_1,X_2)=\alpha^2 + \beta^2 + 2\alpha\beta\rho$, which means $$1=\alpha^2 + \beta^2 + 2\alpha\beta\rho \ (3)$$ instead of (2).

Therefore, you need solve $$0=\alpha+\beta\rho\ (4)$$ $$\alpha^2 + \beta^2 + 2\alpha\beta\rho = 1\ (5)$$

From (4) you get $$\alpha=-\beta\rho\ (6)$$ after squaring you have $$\alpha^2=\beta^2\rho^2\ (7)$$ Put (6) and (7) into (5) $$\beta^2\rho^2+\beta^2-2\beta^2\rho^2=1\ (8)$$ $$\beta^2-\beta^2\rho^2=1\ (9)$$ Therefore $$\beta=\pm\sqrt\frac{1}{1-\rho^2}\ (10)$$ Using (6) and (10), you get $$\alpha=\sqrt\frac{\rho^2}{1-\rho^2},\ \beta=-\sqrt\frac{1}{1-\rho^2}$$ or $$\alpha=-\sqrt\frac{\rho^2}{1-\rho^2},\ \beta=\sqrt\frac{1}{1-\rho^2}$$