Here is a puzzle challenge for you: Suppose $X,Y$ are independent and identically distributed Random Variables. Show that $$P\{|X-Y|\le2\}\le 3P\{|X-Y|\le 1\}$$
2026-05-05 10:43:43.1777977823
An interesting puzzle
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This is known as the 123 Theorem, and a proof can be found here.