I have a set of independent random variables $X_1,...,X_N$ and I know their probability density function $p_X$. In a second step I select one out of the random variables as the minimizer of some function $g$: $Y \in \underset{X \in \{X_1,...,X_N\}}{\arg\,\min} g(X)$. How can I calculate $\operatorname{E}[Y]$ in this general setting, and for what $g, p_X$ would $\operatorname{E}[Y] = \operatorname{E}[X_1]$ hold (if feasible at all)? I am aware that this in a very general question, so even partial answers are welcome (see the simplified formulation below). I am mainly having trouble figuring out the probability density associated with $Y$.
A more specific and less general variant of the problem above, that I am interested in (even though ideally I want to understand how to work with the general case above too) is the following: $N=2,\, g(x) = |x-\alpha|, \alpha \in \mathbb{R},\, \operatorname{supp}(p_X) = [0,1]$, where $\operatorname{supp}$ denotes the support of $p_X$.
Suppose that the function $g$ and $p_X$ are such that $\mathsf{P}(g(X_1)=g(X_2))=0$. Then \begin{align} \mathsf{E}[f(Y)]&=\sum_{i=1}^N\mathsf{E}\!\left[f(X_i)1\!\left\{g(X_i)<\min_{j\ne i}\{g(X_j)\}\right\}\right] \\ &= N\mathsf{E}\!\left[f(X_1)1\!\left\{g(X_1)<\min_{j> 1}\{g(X_j)\}\right\}\right], \end{align} where $f$ is any suitable function. By conditioning on $X_1$, the RHS of the last equations can be written as $$ N\mathsf{E}\!\left[f(X_1)\mathsf{P}(g(X_2)>g(X_1)\mid X_1)^{N-1}\right]. $$ Setting $f(x)=1\{x\le \cdot\}$, you may find the CDF of $Y$.