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suppose I have independent random variables $X_i$ which are distributed binomially via $$X_i \sim \mathrm{Bin}(n_i, p_i)$$.

Are there relatively simple formulae or at least bounds for the distribution $$S = \sum_i X_i$$ avaialble?

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3 Answers

up vote 7 down vote accepted

See this paper (The Distribution of a Sum of Binomial Random Variables by Ken Butler and Michael Stephens).

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I have the same question and i read the paper (The Distribution of a Sum of Binomial Random Variables by Ken Butler and Michael Stephens). unfortunately the approximations are not clear to me ( for example how are the probabilities in Table 2 calculated?) –  May Dec 6 '12 at 22:33
    
@May: I had the same problem these days and I ended up using the explicit formula given in the linked paper. Should be fine if you don't have too many samples. –  Michael Kuhn Dec 7 '12 at 20:26
1  
@MichaelKuhn: so you used poisson binomial distribution function en.wikipedia.org/wiki/Poisson_binomial_distribution, unfortunately I have many samples and I need to use an approximation. –  May Dec 11 '12 at 0:16
    
@May: These seems to be an R package for this distribution: cran.r-project.org/web/packages/poibin/poibin.pdf –  Michael Kuhn Dec 11 '12 at 9:09
    
@MichaelKuhn: thank you. I also found this paper sciencedirect.com/science/article/pii/S0167947312003568# –  May Dec 11 '12 at 15:40
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This answer provides an R implementation of the explicit formula from the paper linked in the accepted answer (The Distribution of a Sum of Binomial Random Variables by Ken Butler and Michael Stephens). (This code can in fact be used to combine any two independent probability distributions):

# explicitly combine two probability distributions, expecting a vector of 
# probabilities (first element = count 0)
combine.distributions <- function(a, b) {

    # because of the following computation, make a matrix with more columns than rows
    if (length(a) < length(b)) {
        t <- a
        a <- b
        b <- t
    }

    # explicitly multiply the probability distributions
    m <- a %*% t(b)

    # initialized the final result, element 1 = count 0
    result <- rep(0, length(a)+length(b)-1)

    # add the probabilities, always adding to the next subsequent slice
    # of the result vector
    for (i in 1:nrow(m)) {
        result[i:(ncol(m)+i-1)] <- result[i:(ncol(m)+i-1)] + m[i,]
    }

    result
}

a <- dbinom(0:1000, 1000, 0.5)
b <- dbinom(0:2000, 2000, 0.9)

ab <- combine.distributions(a, b)
ab.df <- data.frame( N = 0:(length(ab)-1), p = ab)

plot(ab.df$N, ab.df$p, type="l")
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One short answer is that a normal approximation still works well as long as the variance $\sigma^2 = \sum n_i p_i(1-p_i)$ is not too small. Compute the average $\mu = \sum n_i p_i$ and the variance, and approximate $S$ by $N(\mu,\sigma)$.

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Unfortunately, I cannot say anything about the Variance. In what direction would the normal approximation go? –  Lagerbaer Mar 30 '11 at 22:31
    
Do you mean, "is the normal approximation an overestimate or an underestimate?" That depends on the range of values you are considering. Both distributions have total mass $1$. –  Douglas Zare Mar 30 '11 at 23:29
    
I'd be interested in an estimate on the expected value. –  Lagerbaer Mar 31 '11 at 3:03
    
To use a normal approximation, you have to know the mean and variance. (To use the more complicated approximations in the paper PEV cited, you need more information, such as the first 4 moments.) If you don't know the expected value, then what do you know about these binomial summands? –  Douglas Zare Apr 3 '11 at 4:41
    
I know $n_i$ and $p_i$ of each of the summands, and hence I know the expected value and variance of each of the summands. –  Lagerbaer Apr 3 '11 at 4:44
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