  Assume that $X_{\Sigma} \in \mathbb{R}^{n}$ is a centered random vector
following a multivariate normal distribution with positive definite covariance
matrix $\Sigma$. Let $g : \mathbb{R}^{n} \to \mathbb{C}$ be measurable and of
moderate growth, say $|g(x)| \lesssim (1 + |x|)^{N}$. We show that the map
$\Sigma \mapsto \mathbb{E}[g(X_{\Sigma})]$ is smooth, and we derive convenient
expressions for its partial derivatives, in terms of certain expectations
$\mathbb{E}[(\partial^{\alpha}g)(X_{\Sigma})]$ of partial (distributional)
derivatives of $g$. As we discuss, this result can be used to derive bounds for
the expectation $\mathbb{E}[g(X_{\Sigma})]$ of a nonlinear function
$g(X_{\Sigma})$ of a Gaussian random vector $X_{\Sigma}$ with possibly
correlated entries.
  For the case when $g\left(x\right) = g_{1}(x_{1}) \cdots g_{n}(x_{n})$ has
tensor-product structure, the above result is known in the engineering
literature as Price's theorem, originally published in 1958. For dimension $n =
2$, it was generalized in 1964 by McMahon to the general case $g :
\mathbb{R}^{2} \to \mathbb{C}$. Our contribution is to unify these results, and
to give a mathematically fully rigorous proof. Precisely, we consider a
normally distributed random vector $X_{\Sigma} \in \mathbb{R}^{n}$ of arbitrary
dimension $n \in \mathbb{N}$, and we allow the nonlinearity $g$ to be a general
tempered distribution. To this end, we replace the expectation
$\mathbb{E}\left[g(X_{\Sigma})\right]$ by the dual pairing $\left\langle
g,\,\phi_{\Sigma}\right\rangle_{\mathcal{S}',\mathcal{S}}$, where
$\phi_{\Sigma}$ denotes the probability density function of $X_{\Sigma}$.
