Suppose $Z_i$ are independent Bernoulli random variables with differing probabilities $P_i$. Also suppose weights $W_i$ are positive and constant. Let's define the random variable $S$ which is the summation of each weighted $Z_i$ (i.e. $S=\sum_i W_iZ_i$).
The Poisson binomial distribution is a special case for this distribution, where $W_i=1$ for all $i$.
Does anyone have any idea of how to compute the skewness for the distribution of the random variable $S$?