# A lower bound on the expected sum of Bernoulli random variables given a constraint on its distribution

Given a set of Bernoulli random variables $$x_1, \dots, x_n$$ (not necessarily identical) with $$X= \sum_{0, I am intrested in finding a lower-bound for $$\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$$.

• 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? – fedja Jun 17 at 5:39