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?

## 1 Answer

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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