# What is an approximation for Poisson binomial distribution?

I am looking for an approximation for Poisson binomial distribution:

The Poisson binomial distribution is the discrete probability distribution of a sum of $n$ independent Bernoulli trials. you can find its pdf in http://en.wikipedia.org/wiki/Poisson_binomial_distribution

In addition, you can find 2 methods for it in the mentioned link. But when I use the second method (using Fourier Transform), the result would be an imaginary number.

I also wanted to use approximations in the following paper http://statistics.stanford.edu/~ckirby/techreports/ONR/SOL%20ONR%20467.pdf

but the approximations are not clear to me. I would be grateful if somebody explains to me:

1) why do I get imaginary number using the second method ( Fourier transform)?
2) and also for example, how are the probabilities in Table 2 in the paper calculated?

• "In addition, you can find 2 approximations for it in the mentioned link." I don't see approximations in the wikipedia link, those are exact formulas. Dec 11 '12 at 1:55
• @leonbloy: Yes you are right! I edited the question.
– May
Dec 11 '12 at 15:44
• How are we supposed to tell you why you get an imaginary number if you don't tell us how you get it?! Dec 11 '12 at 16:05
• @joriki: For example consider these success probabilities: p1=0.5, p2=0.3, p3=0.7, p4=0.3, p5=0.9. And now we want to calculate P(K=2) . Using The first and second(recursive one) formula (in en.wikipedia.org/wiki/Poisson_binomial_distribution) I get 0.3167, but using the third one (Fourier Transform) I get 0.1957 + 0.0944i.
– May
Dec 11 '12 at 16:32
• As I said, I don't understand how you think we can tell you why you got a wrong result if you don't show us how you got it. The Wikipedia formula is manifestly real-valued, since the contributions for $l$ and $n+1-l$ are complex conjugates of each other, and the remaining contributions for $l=0$, and if $n$ is odd for $l=(n+1)/2$, are real-valued. Dec 11 '12 at 17:48

Your second link does not work, so I am not sure whether you have seen that, but Le Cam's Theorem works quite well for small $p_i$ and large $n$.