I should be able to prove this. But I'm not... I also browsed the web extensively with no success.

Hope someone can show me how to prove it. Thanks in advance!

Given two uncorrelated standard normal distributions X1 and X2, the linear combination

Y = aX1 + bX2

is standard normal correlated to X1 by the coefficient

a/sqrt(a^2+b^2)

It also can be extended to many distributions through the Cholesky decomposition of their covariance matrix.

I would also appreciate very much links to textbooks or web pages that formally prove the concept

Thanks and Regards