  We obtain the von K\'arm\'an-Howarth relation for the stochastically forced
three-dimensional Hall-Vinen-Bekharvich-Khalatnikov (3D HVBK) model of
superfluid turbulence in Helium ($^4$He) by using the generating-functional
approach. We combine direct numerical simulations (DNSs) and analyitcal studies
to show that, in the statistically steady state of homogeneous and isotropic
superfluid turbulence, in the 3D HVBK model, the probability distribution
function (PDF) $P(\gamma)$, of the ratio $\gamma$ of the magnitude of the
normal fluid velocity and superfluid velocity, has power-law tails that scale
as $P(\gamma) \sim \gamma^3$, for $\gamma \ll 1$, and $P(\gamma) \sim
\gamma^{-3}$, for $\gamma \gg 1$. Furthermore, we show that the PDF
$P(\theta)$, of the angle $\theta$ between the normal-fluid velocity and
superfluid velocity exhibits the following power-law behaviors: $P(\theta)\sim
\theta$ for $\theta \ll \theta_*$ and $P(\theta)\sim \theta^{-4}$ for $\theta_*
\ll \theta \ll 1$, where $\theta_*$ is a crossover angle that we estimate. From
our DNSs we obtain energy, energy-flux, and mutual-friction-transfer spectra,
and the longitudinal-structure-function exponents for the normal fluid and the
superfluid, as a function of the temperature $T$, by using the experimentally
determined mutual-friction coefficients for superfluid Helium $^4$He, so our
results are of direct relevance to superfluid turbulence in this system.
