Is the expectation of minimums less than the minimum of expectations: $\mathbb{E}[\min_{k\in [N]} X_k] \leq \min_{k\in [N]} \mathbb{E}[X_k]$?

Question

Is it true that $\mathbb{E}[\min_{k\in [N]} X_k] \leq \min_{k\in [N]} \mathbb{E}[X_k]$ for random variables $X_k$'s (or say when the $X_k$'s are non-negative)?

I am tempted to say Jensen's inequality, but I am a bit unsure about the concavity of minimum.

Answer 1

$\min_k X_k \leq X_k$, so monotonicity implies $E \min_k X_k \leq E X_k$. This is true for every $k$, so $E \min_k X_k \leq \min_k E X_k$.