Sunday, 25 June 2017

probability - show mathbbEvertXvert=inti0nftymathbbP(vertXvert>y)dyleqsumin=0nftymathbbPvertXvert>y.



Looking for a hint to show E|X|=0P({|X|>y})dyn=0P{|X|>y}.



This is from Theorem 2.3.7 in Durrett (Probability: Theory and examples)




The first equality makes sense by Fubini and the definition of expectation (Durrett Lemma 2.2.8). I'm having a hard time showing the second, though. My gut intuition makes me feel like it should be the other direction.


Answer



As you mention, the first equation is essentially Fubini, along with rewriting P{|X|>y} as an integral.



For the inequality on the right: observe that
0P{|X|>y}dy=n=0n+1nP{|X|>y}dyn=0n+1nP{|X|>n}dy=n=0P{|X|>n}
using the fact that for yx, P{|X|>y}P{|X|>x} since {|X|>y}{|X|>x}.


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