Given a set of Bernoulli random variables $x_1, \dots, x_n$ 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 $k=4$.

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 $k=4$.