# Simulation of beta-binomial distribution

1. May 4, 2012

### fchopin

Hi all!

I'm trying to solve the following problem.

The number of successes in a sequence of N yes/no experiments (i.e., N Bernoulli trials), each of which yields success with probability p, is given by the well-known binomial distribution. This is true if the success probability p is constant and the same for all the N trials.

However, when the probability of success, p, is different for each trial, p_1, p_2, ..., p_N, then the number of successes does not follow a binomial distribution, but a Poisson's binomial distribution instead:

wikipedia--> /Poisson_binomial_distribution

I understand that the Poisson's binomial distribution is valid for any set of probabilities p_1, p_2, ..., p_N.

In my problem, I know that the probabilities p_1, p_2, ..., p_N follow a beta distribution. I found out that, in such a case, the resulting PMF of the number of successes in N trials is given by the beta-binomial distribution:

wikipedia --> /Beta-binomial_distribution

However, I have been playing a bit with some simulation and it seems that this distribution does not fit the resulting PMF. I'm attaching a Matlab file that makes some simulation and generates the PMFs.

What am I doing wrong? Is it possible to exploit the knowledge that the p_1, p_2, ..., p_N follow a beta distribution to simply the general Poison's binomial case? What is the PMF that I need?

Many thanks in advance!

Fryderyk C.

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2. May 4, 2012

### Stephen Tashi

I don't know matlab. Did you draw a new value of p on each bernoulli trial? You comment says you intended to, but I don't see where this happened inside your loop on t.

3. May 5, 2012

### fchopin

Thanks for quick reply.

Basically my problem is that I have a set of N "things" I observe at different time instants t, and I know the success/failure probabilities for each "thing". What I need is the probability to observe k successes in each observation of the N "things".

So I first generate a vector p with N beta-distributed random numbers --> p = betarnd(a,b,1,N), which remains always the same.

Then I use the same set of probabilities p = p_1, p_2,...,p_N in every iteration of the loop on t.

In the problem I'm studying, the N values of p are known and constant. I thought this might be the reason why the beta-binomial distribution doesn't fit. However, I tried drawing different values of p in every iteration of the loop on t and the result doesn't fit the beta-binomial distribution either. In this case, even the Poisson's binomial distribution doesn't fit!! Any ideas? Thanks!

4. May 5, 2012

### chiro

If you have independent events for each 'thing' but the probabilities for each thing is different, then you can use what is called a probability generating function to generate the distribution even if each thing has a different probability.

http://en.wikipedia.org/wiki/Probability-generating_function

5. May 9, 2012

### viraltux

I don't think your description fits the Beta-Binomial distribution but rather the Poisson-Binomial distribution, the moment you know the probability of success of your event within a period t it is irrelevant if you know it because X follows a Beta distribution or any other, the fact is that you know it.

Last edited: May 10, 2012
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