Bivariate Gaussian Distribution
A Gaussian pair given by means, variances, and correlation .
Definition 4.49
Let be a Gaussian vector in . Let , and
Then is called a bivariate Gaussian vector with parameters and .
The correlation lies in [-1, 1]
By the Cauchy-Schwarz inequality, . The covariance matrix of is
and for and this matrix is non-negative definite: for any ,
where the first identity shows non-negativity for and the second for .
Conditional expectation is affine
For a bivariate Gaussian vector,
To see this, write with and , so that
Since is a Gaussian vector, its uncorrelated coordinates and are independent, whence while . Therefore
Related
Stated in
- Definition 4.49§4.9.4 Bivariate Gaussian Distribution