Suppose $Z_i$ are independent Bernoulli random variables with differing probabilities $P_i$. Also suppose weights $W_i$ are positive and constant.
Can you tell me the mean and variance for the random variable $S$ which is the summation of each weighted $Z_i$ (i.e. $W_iZ_i$). Furthermore is there a simple distribution for this random variable (similar to the Poisson Binomial distribution)?