Let $X_1, \ldots, X_N$ be $N$ hidden random iid variables, all with the same standard distribution, let's say uniform $\mathcal{U}(0, 1)$ or Gaussians $\mathcal{N}(0, 1)$ (probably easiest). I observe $N$ corresponding 'noisy' variables $Z_n = X_n + \mathcal{N}(0, \sigma^2)$. I know how to derive the distribution of $X^* = \max({X_1, \ldots, X_N})$ (and correspondingly $Z^*$ if the $X$'s are Gaussians). What I would like to know is how to compute the distribution (or at least the expectation) of $X_{\mathrm{argmax}_n(Z_n)}$.
Intuitively if $\sigma^2$ is small my observed variables will closely follow the hidden ones, and the distribution will be close to $X^*$, while if it is big, they will be dominated by the noise, and the distribution will be the original one of the $X$'s.