How can I effectively model the sum of Bernoulli random variables?

Question

I use Perl to model a random variable ( Y), which is the sum of some independent random variables Bernoulli ( X_i), each of which has a different probability of success ( p_i). Formally Y=Sum{X_i}, where Pr(X_i=1)=p_iand Pr(X_i=0)=1-p_i.

I am interested in quickly responding to requests such as Pr(Y<=k)(where k).

I am currently using random simulations to answer such requests. I randomly draw each X_iaccording to it p_i, then summarize all the values X_ito get Y'. I repeat this process several thousand times and return some times Pr(Y'<=k).

Obviously, this is not entirely accurate, although the accuracy increases significantly as the number of simulations they use increases.

Can you imagine a reasonable way to get an accurate probability?

Answer 1

Firstly, I would not use the built-in for this rand, which is too dependent on the implementation of the C base library to be reliable (see, for example, a blog post indicating that the range randon Windows has a capacity of 32,768).

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Answer 2

, ? : http://newsgroups.derkeiler.com/Archive/Sci/sci.stat.consult/2008-05/msg00146.html

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Answer 3

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