Given a set of Bernoulli random variables $x_1, \dots, x_n$ (not necessarily identical) with $X= \sum_{0<i\leq n} x_i$, I am intrested in finding a lower-bound for $\frac{\mathbb{E} [ \min (X,k) ]}{\mathbb{E} [X]}$ in terms of $k$ and $\alpha$ where $\alpha > \Pr[X>k]$. For example, I want to show that this ratio is a large enough constant for $\alpha=0.2$ and $k=4$.
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2$\begingroup$ Assuming independence (otherwise there is nothing to say), the trivial lower bound (which immediately follows from the log-concavity of the sequence $P(X\ge m), m\in \mathbb Z$) is $1-\alpha^{\frac k{k+1}}$. Do you want something tighter than that? $\endgroup$– fedjaCommented Jun 17, 2019 at 5:39
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