  Pulsar timing-array correlation measurements offer an exciting opportunity to
test the nature of gravity in the cosmologically novel nanohertz gravitational
wave regime. The stochastic gravitational wave background is assumed Gaussian
and random, while there are limited pulsar pairs in the sky. This brings
theoretical uncertainties to the correlation measurements, namely the pulsar
variance due to pulsar samplings and the cosmic variance due to Gaussian
signals. We demonstrate a straightforward calculation of the mean and the
variances on the Hellings-Downs correlation relying on a power spectrum
formalism. We keep arbitrary pulsar distances and consider gravitational wave
modes beyond Einstein gravity as well as off the light cone throughout, thereby
presenting the most general and, most importantly, numerically efficient
calculation of the variances.
