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I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows: ​$$...

Let $X$ be a Gaussian vector in $\mathbb{R}^n$ with $\mathbb{E}[X]=0$ and $\mathbb{E}[X X^\intercal]=I_n$. Let $\mathbf{S}$ be a convex polytope in $\mathbb{R}^n$ defined as the intersection of $m$ $(...

Let $Y = (Y_1, Y_2) \sim N(0, 11^T + I)$, be a bivariate normal random variable with non-isotropic covariance. Define $y = (y_1, y_2)$ and let \begin{align} F_{\delta}(y) = \Pr[Y_1 > y_1 - \delta, ...

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...

I have two Gaussian random variables $$X \sim \mathcal N(0, I), \ \ \ W \sim \mathcal N(0, \sigma\cdot I)$$ and I known a parametric change-of-variable $Y(\theta) = T(X; \theta)$. I would like to ...