Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

We have two random variables $X$ and $Y$. Suppose $P_1$ is the probability that $Pr[X > Y]$. $Z_1$ and $Z_2$ are two i.i.d. (identical and independent) random variables, and let $P_2$ be the probability that $Pr[X+Z_1 > Y+Z_2]$.

Can $P_2$ be greater than $P_1$?

If we think $Z_1$ and $Z_2$ as noise, then in words, can adding i.i.d noise make two random variables more separable?

share|improve this question

1 Answer 1

up vote 2 down vote accepted

It seems that you need more assumptions on $X$ and $Y$. Otherwise, take both to be constant and $X = x < Y = y$, so that $P_1 = 0$. Then it is very easy to find examples such that $P_2 > 0$.

However, this does not mean that the two random variables are made more separable...

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.