Assume that $X_{\Sigma}\in\mathbb{R}^{n}$ is a random vector following a multivariate normal distribution with zero mean and positive definite covariance matrix $\Sigma$. Let $g:\mathbb{R}^{n}\to\mathbb{C}$ be measurable and of moderate growth, e.g., $|g(x)| \lesssim (1+|x|)^{N}$. We show that the map $\Sigma\mapsto\mathbb{E}\left[g(X_{\Sigma})\right]$ is smooth, and we derive convenient expressions for its partial derivatives, in terms of certain expectations $\mathbb{E}\left[(\partial^{\alpha}g)(X_{\Sigma})\right]$ of partial (distributional) derivatives of $g$. As we discuss, this result can be used to derive bounds for the expectation $\mathbb{E}\left[g(X_{\Sigma})\right]$ of a nonlinear function $g(X_{\Sigma})$ of a Gaussian random vector $X_{\Sigma}$ with possibly correlated entries. For the case when $g(x) =g_{1}(x_{1})\cdots g_{n}(x_{n})$ has tensor-product structure, the above result is known in the engineering literature as Price's theorem, originally published in 1958. For d 查看全文>>