Originally Posted by

**theodds** Well, it makes sense.

From a statistical point of view, without restrictions on the values that the p_i take, it would be very hard to conduct any kind of meaningful inference about the p_i if they were unknown.

For what it's worth, it seems clear that a sum of binomial random variables produces one of these guys (a Poisson-Binomial, that is), with some restrictions on the values the p_i can take. In the special case where the original binomial random variables all have the same success probability, you just end up with another binomial, which is exactly what you would expect.

For example, if $\displaystyle X$ is distributed $\displaystyle bin(n, p_1)$ and $\displaystyle Y$ is distributed $\displaystyle bin(m, p_2)$ then $\displaystyle X + Y$ will be distributed as a Poisson-Binomial with parameters $\displaystyle (\underbrace{p_1, ..., p_1}_n, \underbrace{p_2, ..., p_2}_m)$.