Suppose we have $N$ indenpendent normal random variables, $X_1, \ldots, X_N$. Suppose $X_i \sim N(0, \sigma_i^2)$ for all $i$. Suppose $X_{(k)}$ is the $k$th order statistic. Then what is $Pr(X_i = X_{(k)})$? In other words, what is the probability that $X_i$ is in the $k$th place from the lowest to the highest of these random variables?
2026-03-25 20:42:06.1774471326
Probability of being in the kth place
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