Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not identical by which I mean that the probability of success from trial to trial varies? How identical and close to independent do they have to be before we see something that resembles a binomial distribution?

share|improve this question

3 Answers 3

up vote 3 down vote accepted

You are asking, I think, when a Central Limit Theorem holds. The simplest form of the CLT is that the binomial distributions Binomial(n,p), suitably rescaled, converge to a normal distribution as n goes to infinity. (This binomial case is usually not called the CLT, but goes under the name of the de Moivre-Laplace theorem.)

Now, a Binomial(n,p) random variable is the sum of n Bernoulli(p) random variables. The usual form of the CLT states that if $S\_n = X\_1 + ... + X\_n$, where the $X\_i$ are independent and identically distributed with mean μ and standard deviation σ, then $(S\_n - \mu n)/(\sigma \sqrt{n})$ converges in distribution to the standard normal as n → ∞.

If the $X\_i$ are in fact dependent, see the link provided by Ori Gurel-Gurevich above.

If the $X\_i$ are independent but not identically distributed, then there are two standard conditions for proving that the rescaled distribution of $S\_n = X\_1 + ... + X\_n$ converges to the standard normal: Lindeberg's condition and Lyapunov's condition. Both are a bit difficult to understand when you first look at them. But the basic idea behind both of them is that if no one of the summands $X\_i$ is too large (in variance) compared to the others, then the normal distribution still appears.

share|improve this answer

Depending on your setting, it might better to consider the asymptotic when $n\rightarrow \infty$, which gives you a Poisson distribution. There are various conditions, weaker than independence, under which Poisson approximation holds. For example, take a look here.

share|improve this answer
    
Also at Pemantle's web page (math.upenn.edu/~pemantle/full-year-webpage.html) are links to two articles by Arratia, Goldstein, and Gordon on which the lecture you linked to is based. –  Michael Lugo Nov 11 '09 at 22:58

The distribution you describe is called a Poisson-binomial distribution. If you do a search with this name you will find a substantial literature on it.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.