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Let $X_i$ be an i.i.d. Bernoulli distributed sequence, with probability $p$ being 1. Now consider an empirical estimation of $p$ with $l$ samples and I am looking for a lower bound for following probability with assumption of $lp>1$

$$ \mathbb{P}\left\{ \frac{1}{l} \sum_{i=1}^l X_i \geq p \right\} $$

The desired lower bound should be independent of $l$ and $p$. My guess is $1/4$.

Note: It's $\geq$, not $>$.

Some background: I am reading Vapnik's "Statistical Learning Theory". Proof of the lemma 4.1 claims that for $lp > 1$, $$ \mathbb{P}\left\{ \frac{1}{l} \sum_{i=1}^l X_i > p \right\} \leq 1/4 $$ However, note that here it is an upper bound and it breaks the proof. I think we should seek for a lower bound the bias probability instead.

Thanks

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