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Let $X_i$ for $i = 1, \ldots, N$ be independent but not necessarily identically-distributed Bernoulli random variables taking values in $\{0,1\}$ where $\mathbb{P}\{X_i = 1\} = p_i$. Define $Y = \sum_{i=1}^N X_i$, so that $Y$ has a Poisson-binomial distribution. Also let $X$ be binomially distributed with parameters $N$ and $p'$ where $p'$ is such that $|p' - 1/2| \leq |p_i - 1/2|$ for all $i \in \{1, \ldots, N\}$.

Now the variance of $X$ is $N(1-p')p'$ and of $Y$ is $\sum_{i=1}^N (1-p_i) p_i$ so it is clear that the variance of $X$ is larger by the assumption on $p'$. I also predict the following to hold for any $\epsilon > 0$ but I am unable to prove it:

$\mathbb{P}\{|Y - \mathbb{E}[Y]| > \epsilon\} \leq \mathbb{P}\{|X - N p'| > \epsilon\}$

Does anyone have any ideas?

Many thanks!

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