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Let $S_n=\tau_1+\cdots+\tau_n$ be a sum of independent Bernoulli random variables such that $\mathbb{P}(\tau_i=1)=p_i$. Is it true that the mode of $S_n$ is either its mean rounded up or rounded down?

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    $\begingroup$ That would be rather surprising, given that there are plenty of values $p_i$ giving the same mean. Have you tried the lazy way to check it (computer simulations)? $\endgroup$
    – fedja
    Nov 22, 2018 at 21:15
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    $\begingroup$ Why does it have to be unimodal at all? $\endgroup$
    – R W
    Nov 22, 2018 at 22:30
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    $\begingroup$ @RW It is strictly log-concave. $\endgroup$
    – fedja
    Nov 22, 2018 at 23:00
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    $\begingroup$ Perhaps you are thinking of Darroch's theorem: if a polynomial with positive coefficients has all of its roots real, then the mean and mode are distant by at most $1$. This applies here: $\prod (1-p_i + p_i x)$ is the Fourier transform, so the answer is yes. See Pitman's survey on log concave (aka strongly unimodal) distributions for a reference to Darroch's result. $\endgroup$ Nov 22, 2018 at 23:44
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    $\begingroup$ Here's the reference. J Pitman. Probabilistic bounds on the coefficients of polynomials with only real zeros. J. Combin. Theory Ser. A, 77(2):279–303, 1997. Darroch's theorem appears somewhere in there; there is a paywall, and I can't get to it. $\endgroup$ Nov 23, 2018 at 0:08

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Darroch's theorem is the following. Let $p = \sum a_i x^i$ be a polynomial with positive coefficients, and suppose that all the roots of $p$ are real (hence negative or zero) [the corresponding distribution of coefficients is called PF, for Polyà frequency]. Then the mean of the distribution $(a_i)$ differs from the mode by less than $1$. [The theorem also gives a partial result on which of the two integers nearest the mean is the mode when the mean is not an integer; but in cases where I wanted to use it, it was just too crude.]

Here the polynomial whose distribution we are interested in is $\prod (1-p_i + p_i x)$, which obviously has only negative real roots. Hence the answer to the question is yes.

I became aware of Darroch's theorem from a preprint of J Pitman, eventually published, Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Combin. Theory Ser. A, 77(2):279–303, 1997 (unfortunately behind an Elsivier paywall), although the preprint might be available somewhere. It contains lots of examples using Darroch's theorem. Fedja found Darroch's paper at projecteuclid.org/euclid.aoms/1177703287.

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