Let's say that I have a bag of unfair coins. Each coin $\mathit{i=1...N}$ has a probability $\mathit{p_i}$ of giving heads when tossed. If I toss each of my coins once and count the number of heads, what distribution (and parameters) will describe the number of heads I get? I know that if I used the same coin $\mathit{i}=I$ the probability mass function would be a binomial distribution $Pr(k;n,p_I) = \frac{n!}{k!(n-k)!}p_I^k(1-p)^{n-k}$ where $\mathit{n}$ describes the number of times I tossed the coin. In this case however I have $\mathit{n}$ different coins with $\mathit{p_1, p_2, ..., p_n}$ probability of heads tossed one time each.
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5Does this answer your question? Sum of Bernoulli random variables with different success probabilities – Maadhav Apr 07 '20 at 06:18