# Proving probability inequality

1. May 8, 2013

### Lily@pie

1. The problem statement, all variables and given/known data

Prove the following
a>0, X is a non-negative function

$Ʃ_{n\in N} P(X>an)≥\frac{1}{a}(E[X]-a)$

$Ʃ_{n\in N} P(X>an)≤\frac{E[X]}{a}$

3. The attempt at a solution

I know that
$\sum_{n\in N} P(X>an)=\sum_{k \in N} kP((k+1)a≥X>ka)=\sum_{k \in N} E[k1_{[(k+1)a,ka)}(X)]$
where $1_{[(k+1)a,ka)}$ is the indicator function.

I'm not sure how to preceed from here.

All I know is that for all ε>0, $E[ε1_{[ε,∞)}(X)]≤E[X]$

Last edited: May 8, 2013
2. May 8, 2013

### haruspex

The second is not true. Is there a factor 1/a missing on the right?

3. May 8, 2013

### Lily@pie

Sorry, my bad... I've edited it...

4. May 8, 2013

### haruspex

You can eliminate a by defining Y = X/a. Does that help?