Let $A \subseteq [0,1]$ be a Borel set and let $B$ be another Borel set such that $B \subseteq A$ and $B = [0,a]$ for some $a \in [0,1]$.
Let $x_1, x_2, \ldots, x_k$ be $k$ i.i.d. random variables distributed uniformly on $A$, and let $y_1, y_2, \ldots, y_k$ be some other $k$ i.i.d. random variables distributed uniformly on $B$.
Finally, let $X = \min\{ x_1, x_2, \ldots, x_k \}$ and $Y = \min\{ y_1, y_2, \ldots, y_k \}$
Intuitively, it seems obvious that $\mathbb{E}[Y] \leq \mathbb{E}[X]$.
What would be a formal and easy reasoning of this inequality?
Best Answer
Since the support of $x_1$ is a superset of $B=[0,a]$, $$ \mathsf{P}(X\ge t)=[\mathsf{P}(x_1\ge t)]^k\ge [\mathsf{P}(y_1\ge t)]^k =\mathsf{P}(Y\ge t), $$ and $$ \mathsf{E}X=\int_0^{\infty}\mathsf{P}(X\ge t)\, dt\ge \int_0^{\infty}\mathsf{P}(Y\ge t)\, dt=\mathsf{E}Y. $$