Covariance and Correlation of Linear Combinations of the Standard Normal Distribution

The standard normal distribution,has mean 0 and variance = standard deviation = 1. If we have two distributionsandthen the covariance between them isIfandare independent thenso If howeverandare linear combinations of independently distributed standard normal distributionssayandthenandare not independent, even thoughandare. We can find theand the correlation betweenand

and similarly

Sinceareare independent,

The formula for the correlation betweenandis

Then the correlation betweenandis

We can follow the same procedure for any linear combination of standard normal distributions.

Ifandthen