- #1
Lily@pie
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Homework Statement
Prove the following
a>0, X is a non-negative function
[itex]Ʃ_{n\in N} P(X>an)≥\frac{1}{a}(E[X]-a)[/itex]
[itex]Ʃ_{n\in N} P(X>an)≤\frac{E[X]}{a}[/itex]
The Attempt at a Solution
I know that
[itex]\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)][/itex]
where [itex]1_{[(k+1)a,ka)}[/itex] is the indicator function.
I'm not sure how to preceed from here.
All I know is that for all ε>0, [itex]E[ε1_{[ε,∞)}(X)]≤E[X][/itex]
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