18.S096: An extra Open Problem

I have just added an extra open problem (4.6.) to the fourth set of lecture notes. I am documenting it here.

Prove or disprove the following conjecture by Feige:

Given n independent random variables X_1,\dots,X_n s.t., for all i, X_i \geq 0 and \mathbb{E} X_i = 1 we have

\mathrm{Prob}\left( \sum_{i=1}^n X_i \geq n+1 \right) \leq 1 - e^{-1}.

See the notes for more information.

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