  This paper is concerned with the statistical properties of the Gram matrix
$\mathbf{W}=\mathbf{H}\mathbf{H}^\dagger$, where $\mathbf{H}$ is a $2\times2$
complex central Gaussian matrix whose elements have arbitrary variances. With
such arbitrary variance profile, this random matrix model fundamentally departs
from classical Wishart models and presents a significant challenge as the
classical analytical toolbox no longer directly applies. We derive new exact
expressions for the distribution of $\mathbf{W}$ and that of its eigenvalues by
means of an explicit parameterization of the group of unitary matrices. Our
results yield remarkably simple expressions, which are further leveraged to
study the outage data rate of a dual-antenna communication system under
different variance profiles.
