# Math Help - Bound probability of sum of two variables

1. ## Bound probability of sum of two variables

If x1 and x2 are independent, Pr(a<x1<b)=p, Pr(c<x2<d)=p, and y=x1+x2. can we prove that Pr(a+c<y<b+d)>= p? I have proved it if x1 and x2 are normally distributed. Is this also true for general distributions?

2. ## Re: Bound probability of sum of two variables

Hello,

Consider $A=\{a and $B=\{a+c

We have that $P(A)=P(a (by independence).

We also know that $\begin{cases} a

Then recall that if we have an implication A => B, then $P(A)\leq P(B)$

This lets us write that Pr(a+c<y<b+d)>= p²

3. ## Re: Bound probability of sum of two variables

Pr(a+c<y<b+d)>= p² is too conservative because p²<p. It can be shown that if x1 and x2 are independently and normally distributed, Pr(a+c<y<b+d)>= p. I am wondering if this is also true for other distributions.

4. ## Re: Bound probability of sum of two variables

Originally Posted by g322
Pr(a+c<y<b+d)>= p² is too conservative because p²<p. It can be shown that if x1 and x2 are independently and normally distributed, Pr(a+c<y<b+d)>= p. I am wondering if this is also true for other distributions.
False in general. Take $X, Y$ to be independent exponential rvs and a = c = 0, b = d = 1.

Moo's work suggests to me that his bound is the best one can expect without more assumptions.

5. ## Re: Bound probability of sum of two variables

Originally Posted by Guy
Moo's work suggests to me that his bound is the best one can expect without more assumptions.
There is no doubt about it ! It's my work your talking about, constantly skimming perfection and exactness.

(hey, I'm just kidding)