Let $ X_1, cdots, X_n $ be identically distributed non-negative real-valued random variables that are *not necessarily independent*. Note that we have trivially for everything $ varepsilon> 0 $:

$$ mathbb {P} left { frac {1} {n} sum_ {i = 1} ^ {n} X_i geq varepsilon right } leq sum_ {i = 1} ^ { n} mathbb {P} {X_i geq varepsilon } = n mathbb {P} {X_1 geq varepsilon }. $$

For $ n geq 3 $, is there still an example where the above inequality holds for equality, or can we get a sharper bound for $ n $? I am particularly interested in the framework where $ n $ is wide.

For $ n = $ 2, the example $ X_1, X_2 sim U [0,1] $ with $ X_2 = 1 – X_1 $ and $ varepsilon = 0.5 $ presents an example where inequality is about equality, but it seems difficult to generalize to $ n> 2 $.