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What is the probability mass function of the sum of $n$ i.i.d categorical variables, $X_i$, each taking values $0,1,2$ with probability $(1-p)^2, 2p(1-p), p^2$, respectively?

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Hint: Your $X_i$ have the same distribution as a binomial random variable with parameters $(2,p)$, and the sum of independent binomial $(n_i,p)$ random variables is a binomial random variable with parameters $(\sum n_i, p)$ – Dilip Sarwate Feb 9 '12 at 17:32
Like @Dilip said, or, for a zero-cleverness solution, use probability generating functions. – Did Feb 9 '12 at 18:52

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