How do I find $P(X_1 > X_2,...X_n)$ if all variables are independent?

Question

Specifically, I have an estimation for all $X$ where $X_i\sim\mathcal{N}\left(\mu_i,\sigma_i\right)$.

Answer 1

You need the variances for any kind of reasonable estimate / bound. The means alone are not enough. In the following let $E$ be your desired event $X_1 > X_2, \dots, X_n$.

E.g. say $\mu_i = 1 + \mu_{i+1}$ i.e. they are already ranked and the gaps are all size $1$. Consider these scenarios:

• If all $\sigma_i \ll 1$ then the means dominate and the distributions are well separated: $P(E) \approx 1$.

• If all $\sigma_i \gg n$ then the means don't really matter and all $n$ distributions overlap a lot: $P(E) \approx 1/n$ by symmetry.

• If $\sigma_1 \gg n$ and all other $\sigma_i \ll 1$ then it really just depends on what $X_1$ is doing: $P(E) \approx 1/2$ because if $X_1 > \mu_1$ then $E$ is very likely, while if $X_1 < \mu_1$ then $E$ is very unlikely.