Imagine we have $k$ Gaussian RVs $$ X_i \sim N(\mu_i, \sigma_i^2) \text{ for } i=1, \ldots, k $$ and we sample from each of them independently to produce a vector, $\vec{x} = (x_1, \ldots, x_k)$.

For one of the Gaussian RVs, say $X_j$, I am interested in computing the probability that it exceeds all others, i.e. $$ \Pr\left\{ \cap_{i\not= j} \, X_j > X_i \right\}. $$

I know I can use Monte Carlo sampling to estimate this probability. But are there any closed-form analytical methods or approximations?