  Consider the centered Gaussian vector $X$ in $\R^n$ with covariance matrix $
\Sigma.$ Randomize $\Sigma$ such that $ \Sigma^{-1}$ has a Wishart distribution
with shape parameter $p>(n-1)/2$ and mean $p\sigma.$ We compute the density
$f_{p,\sigma}$ of $X$ as well as the Fisher information $I_p(\sigma)$ of the
model $(f_{p,\sigma} )$ when $\sigma $ is the parameter. For using the
Cram\'er-Rao inequality, we also compute the inverse of $I_p(\sigma)$. The
important point of this note is the fact that this inverse is a linear
combination of two simple operators on the space of symmetric matrices, namely
$\P(\sigma)(s)=\sigma s \sigma$ and $(\sigma\otimes \sigma)(s)=\sigma \,
\mathrm{trace}(\sigma s)$. The Fisher information itself is a linear
combination $\P(\sigma^{-1})$ and $\sigma^{-1}\otimes \sigma^{-1}.$ Finally, by
randomizing $\sigma $ itself, we make explicit the minoration of the second
moments of an estimator of $\sigma$ by the Van Trees inequality: here again,
linear combinations of $\P(u)$ and $u\otimes u$ appear in the results.
