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)?
Best Answer
I understand $Z_i$ takes value 1 with probability $P_i$ and $0$ otherwise.
Computing the mean of the weighted sum is simple: by linearity of the $\mathrm E$ operator,
$$ \mathrm E \sum_i W_i Z_i = \sum_i W_i \mathrm E Z_i = \sum_i W_i P_i. $$
As for the variance, since the variables are independent you can just sum their variances. So:
$$ \mathrm{Var} \sum_i W_i Z_i = \sum_i W_i^2 \mathrm{Var}Z_i = \sum_i W_i^2 P_i(1-P_i). $$
The distribution of $\sum_i W_i Z_i$ does not have a name, to my knowledge.