TITLE: Sum of two random variable X and -X
QUESTION [1 upvotes]: Let $X$ and $Y$ be two random variables such that $X$ and $-Y$ have the same distribution. Prove that $P(X+Y<-t)=P(X+Y>t)$. I know how to prove this when $X$ and $Y$ are independent but how do I prove this for when they are dependent?

REPLY [0 votes]: If $X\sim -Y$ then $X+Y \sim -Y-X$.   It follows that $\mathsf P(X+Y < - t) = \mathsf P(-Y-X < - t)$.
$$\therefore \mathsf P(X+Y< -t) = \mathsf P(X+Y> t)$$
Basically, if $X$ and $-Y$ are distributed identically then the sum $X+Y$ is distributed symmetrically about $0$.
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