Bounds on convex combination of two random variables.

Question

A random vector $(X,Y)$ defined on the probability space $(\Omega, \mathcal F, \mathbb P)$. For any $\omega \in \Omega$, we have $X\leq Y$. Both $X$ and $Y$ are bounded continuous random variable. Suppose $\mathbb F_Y$ is the cumulative distribution function (cdf) of $Y$, and $\mathbb F_X$ is the cdf of $X$. We further define $Z_\lambda=(1-\lambda)X+\lambda Y$, $\lambda \in [0,1]$. How to prove $$\inf_{\lambda\in [0,1]} \mathbb P (Z_\lambda \in [l,u]) = \min\{\mathbb P (X \in [l,u]),\mathbb P (Y\in [l,u])\},$$ where $l=F_X^{-1}(t)$, $u=F_Y^{-1}(1-t)$ , $l< u$?

Answer 1

This is false. Take $X\sim \frac{\delta_2+\delta_0}{2}$, $Y=X+2$, and $[l,u]=[1.5,2.5]$.