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Let $X_1, X_2, ... X_n$ be independently drawn from different normal distributions, such that $ X_i \sim N(\mu_i, \sigma^2_i) $

For any $j$ what is the probability that $X_j$ is the greater than all other $X$?

E.g. for $j=1$ what is $P(X_1 > X_2 \land X_1 > X_3\land ...\land X_1 > X_{n})?$ E.g. when $n=3$, what is $P(X_1 > X_2 \land X_1 > X_2)?$

Initially I incorrectly thought $P(X_1 > X_2 \land X_1 > X_2) = P(X_1 > X_2) *P(X_1>X_3)$. However, if we imagine all distributions are equal, then $P(X_i > X_j) = 0.5$ so the product becomes $0.5 * 0.5 = 0.25$ however in reality $P(X_j \textrm{ is greatest) = 1/3}$ when $n=3$.

I'm currently running numerical simulations to calculate this probability as part of a multi-armed-bandit simulation project I'm working on, but I feel that there must be an analytical solution to this I can use to save compute time?

Andreas
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    This has been asked several times here on CV and never answered because there is no analytical solution: you need to perform numerical integration. – whuber Aug 17 '22 at 16:25
  • ... or simulation – Henry Aug 17 '22 at 18:01
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    See https://stats.stackexchange.com/questions/44139/what-is-px-1x-2-x-1x-3-x-1x-n and the questions linked to it – Henry Aug 17 '22 at 18:04
  • I should add that there is an analytical solution for $n=2$ (even when the variables are correlated), as reported at https://stats.stackexchange.com/a/579392/919. The method used there (as illustrated at https://stats.stackexchange.com/a/139090/919) indicates what the problem is: with $n=2$ we can obtain a solution in terms of distribution functions of $n-1=1$-variate Normals; but with $n \gt 2,$ we need to compute the distribution functions of multivariate Normals. – whuber Aug 17 '22 at 18:55

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