# Is the mode of a Poisson Binomial distribution next to the mean?

A Poisson-Binomial variable $$X∼PB(p1,…,pn)X\sim PB(p_1, \dots, p_n)$$ is the sum of $$nn$$ independent, not necessarily identically distributed, Bernoulli variables $$X1,…,XnX_1, \dots, X_n$$:
$$X=n∑i=1Xi, X=\sum_{i=1}^n X_i,$$
with $$Xi∼Ber(pi)X_i\sim Ber(p_i)$$.

The Poisson-Binomial distribution is unimodal and $$E[X]=∑ni=1pi=μ.\mathbb E[X]=\sum_{i=1}^np_i=\mu.$$

Question: is it always the case that the mode is either $$⌊μ⌋\lfloor \mu \rfloor$$ or $$⌈μ⌉\lceil \mu \rceil$$?

It seems to me that this is the case. The PB distribution is a generalization of the Binomial distribution, and has lower or equal entropy. The probability mass is thus more concentrated around the mean in some sense, which suggests that the answer is yes.

I have done some numerical experiments and have not found a counterexample, which reinforces my suspicion.

$$mode={kifk≤μ≤k+1k+2,kork+1ifk+1k+2≤μ≤k+1−1n−k+1,k+1ifk+1−1n−k+1≤μ≤k+1.\begin{equation} mode= \begin{cases} k \hspace{3mm}if\hspace{3mm}k \leq \mu \leq k+\frac{1}{k+2},\\ k \hspace{3mm}or\hspace{3mm} k+1\hspace{3mm}if\hspace{3mm} k+\frac{1}{k+2} \leq \mu \leq k+1 - \frac{1}{n-k+1},\\ k+1\hspace{3mm}if\hspace{3mm}k+1 - \frac{1}{n-k+1} \leq \mu \leq k+1. \end{cases} \end{equation}$$
Consequently, the mode differs from the mean by at most $$11$$. Note that a Poisson binomial distribution can have either one or two consecutive modes.