CDF of minimum of correlated and iid random variables

Question

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are independent $W_3$, $W_4$ are correlated. We assume that the range of both $X_1$ and $X_2$ is from $0$ to $1$. Is it true that

$$\Pr \{X_2 \leq x \} \leq \Pr \{X_1 \leq x \},~~~ \forall x?$$

If yes, how to prove it? I would be grateful if any pointers to existing literature are given. Thanks!

Answer 1

$$ \begin{eqnarray*} P\left(X_2> x\right)&{}={}&P\left(W_3> x,\ W_4> x\right)\newline &{}={}&\mathbb{E}\left[{\bf{1}}_{\left\{W_3> x,\ W_4> x\right\}}\right]\newline &{}={}&\mathbb{E}\left[{\bf{1}}_{\left\{W_3> x\right\}}{\bf{1}}_{\left\{W_4> x\right\}}\right]\newline &{}={}&\mathbb{E}\left[{\bf{1}}_{\left\{W_3> x\right\}}\right]\mathbb{E}\left[{\bf{1}}_{\left\{W_4> x\right\}}\right]{}+{}\mathbb{C}ov\left({\bf{1}}_{\left\{W_3> x\right\}},\ {\bf{1}}_{\left\{W_4> x\right\}}\right)\newline &{}={}&P\left(W_1> x\right)P\left(W_2> x\right){}+{}\mathbb{C}ov\left({\bf{1}}_{\left\{W_3> x\right\}},\ {\bf{1}}_{\left\{W_4> x\right\}}\right)\newline &{}={}&P\left(X_1> x\right){}+{}\mathbb{C}ov\left({\bf{1}}_{\left\{W_3> x\right\}},\ {\bf{1}}_{\left\{W_4\ge x\right\}}\right)\newline \end{eqnarray*} $$

Therefore,

$$ \begin{eqnarray*} \mathbb{C}ov\left({\bf{1}}_{\left\{W_3> x\right\}},\ {\bf{1}}_{\left\{W_4> x\right\}}\right){}\ge0&\iff& P\left(X_2> x\right)\ge P\left(X_1> x\right)\newline &&\newline &\iff& P\left(X_2\le x\right)\le P\left(X_1\le x\right)\,. \end{eqnarray*} $$