Let xi be independent Bernoulli random variables with success probabilities pi. That is, xi=1 with probability pi and xi=0 with probability 1−pi.
Is there a closed expression or an approximate formula for the distribution of the sum ∑ixi?
Yes, in fact, the distribution is known as the Poisson binomial distribution, which is a generalization of the binomial distribution. The distribution’s mean and variance are intuitive and are given by
E\left[\sum_i x_i\right] &= \sum_i E[x_i] = \sum_i p_i\\
V\left[\sum_i x_i\right] &= \sum_i V[x_i] = \sum_i p_i(1-p_i).
The expectation is straightforward because it is a linear operator. The variance is also straightforward because of the independence assumption.