I have a list of N invitees to whom I have sent the invitation. I know the individual probabilities of each invitee attending the wedding. I would like to calculate the number of invitees who are likely to attend the wedding. How do I calculate this?
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1By likely do you mean most probable value, or expected value? – Dec 07 '13 at 04:20
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When you say "how do I calculate this?", what do you mean, exactly? – Igor Rivin Dec 07 '13 at 04:24
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We can treat each invitee as a Bernoulli distribution and the number of invitees who attend as the sum of $N$ of such Bernoulli distributions. Assuming independence, we can get the expectation by $ E(X)=\displaystyle \sum_i^N p_{i=1}$ (i.e. sum of individual probabilities).
If we are interested in the most probable value, (i.e. the mode), I think more analysis will be needed. If all the individual probabilities are the same then the number of invitees attending will follow a Binomial distribution and the mode is known for such cases (see here). If the probabilities are not the same, I do not know of a simple method. (Only one that comes to mind is getting the probability distribution function by brute force (counting over all possible subsets) and then proceeding to maximize the probability.)
Kelvin Soh
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1Kevin is correct that the general PMF is not nice to calculate. See this related thread: http://math.stackexchange.com/questions/392860/sum-of-bernoulli-random-variables-with-different-success-probabilities. If you are looking for an approximate answer, you can try using the BetaBinomial distribution, where you fit a beta distribution to the distribution of acceptance probabilities. That might get you a rough estimate. – Dec 07 '13 at 05:29