The Poisson-Binomial distribution concerns the probability of the number of successes in a sequence of independent Bernoulli trials with different probabilities of success. This is a generalization of the Binomial distribution.
With the command dpoibin, in the poibin package, one can obtain the mass probability function. For instance, with this command:
library(poibin)
n <- 100
Probs_Success <- runif(n)
dpoibin(kk = 30, pp = Probs_Success)
one can obtain the probability of getting 30 successes in a sequence of 100 independent Bernoulli trials with probabilities of success contained in the vector Probs_Success. To compute this probability one has to sum the probabilities of all possible sequences of lenght 100 where there are 30 successes and 70 failures.
Question: How can I efficiently obtain all of the summands that generate the above probability in R? Thanks a lot for your help.
For those who are interested in the problem that motivated this question, please click on the following link:
https://math.stackexchange.com/questions/2924831/bivariate-poisson-binomial-distribution
I solved the problem that motivated the above question. This solution is based on the paper
https://www.tandfonline.com/doi/abs/10.1080/03610918708812585
To see this solution, just click on the following link
https://math.stackexchange.com/questions/2924831/bivariate-poisson-binomial-distribution
Thank all people who provided me suggestions.