# Probability that one Gaussian RV exceeds all others (not the identically distributed case)

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?

• Do you mean $Pr(\{X_j > \max_{i \not= j \colon 1 \leq i \leq k} ~X_i\})$? – Dieter Kadelka Jan 24 at 10:37
• Yes, that's another way to write it. The intersection of the events "$X_j > X_i$" for all $i$ is equivalent to $X_j > \text{max}(\{X_i\}_{i\not=j})$. – ted Jan 24 at 17:31
• this is answered at stats.stackexchange.com/a/114519 – Carlo Beenakker Jan 24 at 19:37