Say we have a collection of independent random variables $X_0, ... , X_n$, where $X_0 \sim \mathcal{N}(\mu_0, \sigma_0^2)$, but $X_i \sim \mathcal{N}(\mu, \sigma^2)$, for $1 \leq i \leq n$. We characterize the distribution of $R_0 := 1 + \sum_{i=1}^{n} \mathbf{1}\{X_i \leq X_0\}$, the rank of the random variable whose distribution potentially differs from that of the others -- the odd normal out. We show that $R_0 - 1$ is approximately beta-binomial, an approximation that becomes equality as $\sigma/\sigma_0$ or $(\mu-\mu_0)/\sigma_0$ become large or small. The intra-class correlation of the approximating beta-binomial depends on $\Pr(X_1 \leq X_0)$ and $\Pr(X_1 \leq X_0, X_2 \leq X_0)$. Our approach relies on the conjugacy of the beta distribution for the binomial: $\Phi((X_0-\mu)/\sigma)$ is approximately $\mathrm{Beta}(\alpha(\sigma/\sigma_0, (\mu-\mu_0)/\sigma_0), \beta(\sigma/\sigma_0, (\mu-\mu_0)/\sigma_0))$ for functions $\alpha, \beta > 0$. We study the distributions of the in-normal ranks. Throughout, simulations corroborate the formulae we derive.

翻译：假设有一组独立随机变量 $X_0, \ldots, X_n$，其中 $X_0 \sim \mathcal{N}(\mu_0, \sigma_0^2)$，而 $X_i \sim \mathcal{N}(\mu, \sigma^2)$（对于 $1 \leq i \leq n$）。我们刻画了 $R_0 := 1 + \sum_{i=1}^{n} \mathbf{1}\{X_i \leq X_0\}$ 的分布，即潜在分布与其他变量不同的随机变量——异常正态变量——的秩。我们证明 $R_0 - 1$ 近似服从贝塔-二项分布，且当 $\sigma/\sigma_0$ 或 $(\mu-\mu_0)/\sigma_0$ 趋于极大或极小时，该近似成为精确分布。近似贝塔-二项分布的组内相关系数取决于 $\Pr(X_1 \leq X_0)$ 和 $\Pr(X_1 \leq X_0, X_2 \leq X_0)$。我们的方法依赖于贝塔分布对二项分布的共轭性：对于函数 $\alpha, \beta > 0$，有 $\Phi((X_0-\mu)/\sigma)$ 近似服从 $\mathrm{Beta}(\alpha(\sigma/\sigma_0, (\mu-\mu_0)/\sigma_0), \beta(\sigma/\sigma_0, (\mu-\mu_0)/\sigma_0))$。我们研究了正常变量内部秩的分布。全文通过模拟验证了所推导的公式。