Posted on Math Stack Exchange, where it received little traction. https://math.stackexchange.com/questions/4385830/effect-of-conditioning-on-quantiles
Suppose that we have three continuous, independent, non-negative random variables $X,Y,Z$. Fix $q\in(0,1)$ and suppose that $$q=\mathbb{P}(X+Y\leqslant \tau)=\mathbb{P}(X+Y+Z\leqslant \pi).$$
For $\mu>0$ such that $\mathbb{P}(X\leqslant \mu)>0$, is it true that $$\mathbb{P}(X+Y\leqslant \tau|X\leqslant \mu) \geqslant \mathbb{P}(X+Y+Z\leqslant \pi|X\leqslant \mu)?$$
In other words, what effect does replacing the unconditional sums $X+Y, X+Y+Z$ with $(X+Y)|(X\leqslant \mu)$, $(X+Y+Z)|(X\leqslant \mu)$ have on the quantiles?
I have tried many simulations with a number of different distributions, and I have yet to see the inequality violated.
I have an intuition for why this should be true, which is that limiting the ability of $X$ to contribute to the sums will make it more difficult for both $X+Y$ and $X+Y+Z$ to exceed their respective thresholds $\tau$ and $\pi$, but the effect of limiting how large $X$ can be will have a stronger prohibitive effect on $X+Y$ than on $X+Y+Z$, because the latter sum has $Z$ to draw from.
