Modified Poisson binomial distribution In the Poisson binomial distribution each trial is either yes (1) or no (0). Is there a distribution where each trial is either yes (N) or no (0)?  
I'd like to model a situation where I have, for example, 10 workers. Each worker can produce $N$ units of good in one day or fail with known probability and produce nothing. $N$ can be different for each worker. I need to know distribution of goods after a day of work. Poisson binomial distribution gives me an answer in case each worker produces 1 good.
 A: First, you probably don't want to use a Poisson distribution, as it describes a random variable that can take on any of an infinite collection of integers. Stick with a Binomial(1,p), otherwise known as a Bernoulli(p).
If $X \sim \text{Bernoulli}(p)$, and $Y = NX$, then the pmf of $Y$ would be 
$$
p_Y(y) = p_X([y/N]) = p^{y/N}(1-p)^{1-y/N}.
$$
For an individual worker, you can simply scale a Bernoulli rv to get what you want.
For adding the outcomes of multiple, independent workers, it is helpful to look at MGFs. The MGF from above is 
$$
M_Y(t) = E[e^{tY}] = E[e^{tNX}] = M_X(tN) = (1-p+pe^{tN}).
$$
If you have several ($M$) workers, all with the same $N$, then the MGF of the sum is
$$
(1-p+pe^{tN})^M.
$$
This is the MGF of the random variable $\sum_{i=1}^M N X_i$. Add together $M$ Bernoulli random variables, and then multiply the result by $N$.
As @jbowman was saying, if you have a different $N_i$ for each worker, then you will not get the above scaled Bernoulli distribution, and you will have to calculate probabilities by hand.
A: This is a Generalized Poisson Binomial distribution. Have a look at R package PoissonBinomial (vignete) implementing a number of new high-performance algorithms for this distribution.
