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I found this question while browsing through "The Probabilistic Method", by Noga Elon.

Let X and Y be 2 independent and identically distributed real valued random variables. Prove that: $$P(|X-Y| \leq 2) \leq 3P(|X-Y| \leq 1)$$

So I tried the following:

$$P\{|X-Y| \leq 2\} = P\{|X-Y| \leq 1\} + P\{X-Y \in (1,2]\cup[-2,-1)\}$$ $$= P\{|X-Y| \leq 1\} + P\{X-Y \in (1,2]\} + P\{X-Y \in [-2,-1)\}$$ $$= P\{|X-Y| \leq 1\} + 2P\{X-Y \in (1,2]\}$$ where the last step follows because $X-Y$ has a symmetric distribution. NB: A random variable Z has symmetric distribution if $$P(Z \leq z) = P(Z \geq -z) \quad \forall z \in \mathbb{R}$$

Thus the problem boils down to showing $$P\{X-Y \in (1,2]\} \leq P(|X-Y| \leq 1)$$ and I would be done. Unfortunately, I don't know how to proceed from here. I appreciate any help, hints, useful comments etc. I receive.

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What does iid mean? – Dominik Nov 25 '12 at 16:45
independent and identically distributed. I'll edit it. – Gautam Shenoy Nov 25 '12 at 16:46
up vote 3 down vote accepted

A proof is given in "The 123 Theorem and its extensions" by Noga Alon, Raphael Yuster. (See also this question.)

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The '123 theorem' looks so simple. The solution seems really wicked! I have been trying this problem for four days now :( – Isomorphism Nov 26 '12 at 7:29

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