# Mode of a sum of Bernoulli random variables

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?

• 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)? Nov 22, 2018 at 21:15
• Why does it have to be unimodal at all?
– R W
Nov 22, 2018 at 22:30
• @RW It is strictly log-concave. Nov 22, 2018 at 23:00
• 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. Nov 22, 2018 at 23:44
• 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. Nov 23, 2018 at 0:08

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.