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Since a Poisson binomial distributed variable is a sum of n independent Bernoulli distributed variables, its mean and variance will simply be sums of the mean and variance of the n Bernoulli distributions:
Probability mass function
The probability of having k successful trials out of a total of n can be written as the sum
[1]
where is the set of all subsets of k integers that can be selected from {1,2,3,...,n}. For example, if n=3, then . is the complement of , i.e. .
will contain elements, the sum over which is infeasible to compute in practice unless the number of trials n is small (e.g. if n=30, contains over 1020 elements). There are luckily more efficient ways to calculate .
As long as none of the success probabilities are equal to one, one can calculate the probability of n successes using the recursive formula
[2]
^
Wang, Y. H. (1993). "On the number of successes in independent trials". Statistica Sinica. 3 (2): 295–312.
^
Chen, X. H (1994). "Weighted finite population sampling to maximize entropy". Biometrika. 81 (3): 457. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
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Fernandez, M. (2010). "Closed-Form Expression for the Poisson-Binomial Probability Density Function". IEEE Transactions on Aerospace Electronic Systems. 46: 803–817. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)