Order statistics: Probability that a normal random variable is k-th out of n when ordered

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 $n$.

• 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 $R$ – conjectures Nov 13 '13 at 8:54