# Bounds on convex combination of two random variables.

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 $$l=F_X^{-1}(t)$$, $$u=F_Y^{-1}(1-t)$$ , $$l< u$$?

• Welcome to the site! You'll attract more answerers if you show what your thinking so far is, and where your efforts have been unsuccessful. Bare problem statements like this often attract downvotes – postmortes Mar 7 at 7:25
• Are you sure this is true ? – Pierre Mar 7 at 13:00

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