Let $X$ and $Y$ be i.i.d. random variables with support $[0,\infty)$. Is the function $f : (0,\infty) \rightarrow \mathbb R$ given by \begin{equation*} f(x) = \Pr(X+Y \le x | X \le x) \end{equation*} increasing?
2026-03-25 22:05:22.1774476322
If X and Y are i.i.d and positive, is Pr(X + Y < x|X < x) increasing in x?
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It is not. Consider $X$ and $Y$ concentrated at $2$ and $6$ with equal probability. Then
$$f(5)=\frac12\gt\frac14=f(7)\;.$$