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My question is almost the same as that of Order statistics: probability random variable is k-th out of n when ordered

with the exception that the underlying distribution from which $X_1$ is drawn, having CDF $F$, is $N(\mu_1,\sigma_1^2)$. And the distribution from which $X_2,X_3,\ldots,X_n$ are drawn, having CDF $G$, is $N(\mu_2,\sigma_2^2)$. If that is the case then how to calculate the probability that $X_1$ is the $k^{th}$ out of the $n$ random variables when they are sorted, in order.

Even if the probability might not be expressible by any elementary formula for large $n$ could you please give hints as to how to numerically approximate. Also, in that case, could you please solve for the case of some small $k$ and $n$.

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  • $\begingroup$ If a numerical answer is OK: - Generate large $R$ vectors $V$ where $V_i \sim N(\mu_i,\sigma^2_i)$. - Sort the data, retaining a copy of the the original indices. - The probability that $V_i$ is the $i^{th}$ order statistic is the number of simulations in which $V_i$ was indeed the $i^{th}$ order statistic, divided by $R$ $\endgroup$ – conjectures Nov 13 '13 at 8:54
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\begin{align}\varrho &= \binom{n-1}{k-1}\mathbb{E}^{X_1}[\mathbb{P}(X_2\le X_1|X_1)^{k-1}\mathbb{P}(X_2\le X_1|X_1)^{n-k}]\\ &=\binom{n-1}{k-1}\int \Phi(\{x_1-\mu_2\}/\sigma_2)^{k-1}\Phi(-\{x_1-\mu_2\}/\sigma_2)^{n-k}\sigma_1^{-1}\varphi(\{x_1-\mu_1\}/\sigma_1)\text{d}x_1\end{align}

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