Calculate the probability of the sum of multiple discrete independent variables

Question

There are $N$ independent discrete variables $\{x_i\}$, $i=1,2,\cdots,N$, where $\Pr(x_i=1)=p_i$ and $\Pr(x_i=0)=1-p_i$.

Let $K=\sum_{i=1}^{N}x_i$. Then, how about the probability $\Pr(K=k)$?

Answer 1

Your initial random variable are independent Bernouli r.vs. So their sum results in the Binomial distribution, $K$ ~ $Binomial(N, p)$, and so $Pr(K=k) = \binom{N}{k} p^k(1-p)^{N-k}$.

Answer 2

Let $p_i$ be the probability of $i^{th}$ coin landing Heads. We want the probability of $k$ coins landing heads. If these were the first $k$ coins,

$$Probability=p_1p_2...p_k(1-p_{k+1})(1-p_{k+2})..(1-p_N)$$

This is just one way of selection of $\binom{N}{k}$ ways. We need to sum over them all to get the total probability. I don't think a simple closed form expression exists for the sum, but here's a try:

$$\sum_{\sum x_i=k}\prod_{i=1}^{N}{p_i}^{x_i}(1-p_i)^{(1-x_i)}$$