Suppose random variables $X_1$, $X_2$,..., and $X_n$ are independent and normally distributed (with varied means and variances), what is $P(X_i\ge \max_{j\in [1,n], j\neq i} \{X_j\})$?
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Assuming that the $X_i$ all have the same distribution, it's $1/n$ as above. Do you mean for it to be generic normal distributions, with possibly differing means and variances? – Danica Mar 14 '18 at 17:03
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@Dougal in my case, the means and variances are varying. – Guoyang Qin Mar 14 '18 at 17:09
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In that case, see https://stats.stackexchange.com/questions/4138/what-is-the-probability-that-random-variable-x-1-is-maximum-of-random-vector, where in your case $\Sigma$ will be diagonal, though maybe that leads to a simpler closed-form case. – Danica Mar 14 '18 at 17:11
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@Dougal the link is what I am looking for, thx. But I still got a question if a multivariate normal distribution with a diagonal sigma matrix is equivalent to multiple independent univariate normal distributions? – Guoyang Qin Mar 14 '18 at 17:27
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[Yes](https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Normally_distributed_and_independent). You can prove it by just writing out the pdf for a diagonal-$\Sigma$ multivariate normal and observing that it factors into exactly the product of independent univariate normals. – Danica Mar 14 '18 at 17:29