Let $(X_{k})$ be a vector of $n$ independent continuous random variables each following a distribution $\mathcal{N}(\mu,\sigma^2)$. If I know the outcome of exactly one of these variables, for example I know that $\mathbb{P}(\bigcup_{k=1}^{n}X_k\in[a-\delta;a+\delta])=1$ for $\delta$ as small as desired, how would I get the expected value of the realization of $(X_{k})$ immediately following $a$ (when sorted in ascending order)?
Intuitively, I would take it to be: $\sum_{k=1}^{n}\mathbb{P}(X_{(k)} \in [a-\delta;a+\delta])E(X_{(k+1)}|X_{(k)}\in[a-\delta;a+\delta])$
(where $X_{(k)}$ is the $k-th$ order of $(X_k)$)
But I have no idea how to formally prove it.
Could you please give me any hint on how to formalize this problem?