# The unimodality of a generalised binomial distribution

In my research, I'm summing a bunch of non-independent, not identically distributed random (0,1)-variables $X_1,X_2,\ldots,X_n$ with corresponding probabilities $\pi_1,\pi_2,\ldots,\pi_n$. The resulting distribution is multimodal, and I want to show that the multimodality is caused by the dependence.

In an attempt to show this, I'm assuming that the probabilities are independent, and want to show that the resulting distribution is unimodal (which I'm not 100% sure is true at this point, although some computational results suggest it is true).

So, the distribution is given by $\mathrm{Pr}[\sum X_i=r] = \sum_{C \subseteq [n]:|C|=r} \big( \prod_{i \in C} \pi_i \big) \big( \prod_{i \not\in C} (1-\pi_i) \big).$

This is a generalised binomial distribution, (we retrieve the binomial distribution in the special case of $\pi_1=\pi_2=\cdots=\pi_n$)

I find it hard to believe that this has not be studied previously, which leads me to:

Question: Does this distribution have a special name I can search for? What are some references where I can find this distribution?

I found quite a few different generalised binomial distributions via Google, but not one that seems to match the above.

Or if you happen to know the answer to the underlying question:

Question: Is the distribution unimodal?