# How can I find distribution from mean and variance

If I know the mean and variance of a discrete random variable $\in \{0,1,\dots \}$, How can I find the distribution?

A mean of 0.6 and a variance of 2.64 could be a negative binomial distribution with parameters $p=0.914$ and $r=0.057$. However, it could just as well be a zero-inflated Poisson distribution or in fact any number of other distributions (just assign probabilities to the realizations in $\{0, 1, 2, \dots\}$ that fulfill the mean and variance requirements), so I would not speak of finding "the distribution".

As others pointed out, you can't go further without any model assumption. Just assume the closest model based on your insight from the data and proceed with a formal statistical inference procedure, for example, possibly, the method of moments in your problem. Remember, all models are wrong, some are useful (George Box).

You can't really find anything out unless you add extra assumptions - in particular, what class of distribution it is. For example if you assume that it's a binomial distribution, the mean is np and the variance is np(1-p), so you can work out the values of the coefficients n and p.

If you have more than one possible distribution to explain the data, you need to search for "model selection" techniques.

• I don't know the class of distribution. Is there any general approach? Binomial distribution does not work for some mean and variances (0.6,2.64 for example). Commented Feb 5, 2013 at 16:23
• You can exclude some distributions (as @Masood_mj shows), but cannot narrow the possible distributions down to one. Commented Feb 5, 2013 at 17:04

What the other answers already suggest is to compare mean and variance.

1. If mean equals variance then it could be Poisson
2. If mean is less than veriaance then it could be negative binomial
3. If mean is greater than variance then it could be binomial

And then there are the inflated and truncated families. So you have to know whether 0 has a special role and whether the distribution has finite support (like Binomial) or infinite (like Poisson and negative binomial).

In fact just knowing mean and variance is not much more than guessing. Do you know anything of the generation of the data (like number of car accidents or number school drop-outs)?

• Can the parameter n in binomial be real? Commented Feb 5, 2013 at 21:50
• I don't know whether such a generalization exists, but in the natural setting $n$ is an integer greater $0$ en.wikipedia.org/wiki/Binomial_distribution Commented Feb 6, 2013 at 8:58

Depends on the size of your support. In this case that we are talking about a discrete support, a distribution is fully specified by a vector with the same size as the support. The distribution, under some technical conditions, can be uniquely mapped to a set of moments, some times referred to as the probability generation function. For this mapping to work, you need all the moments though. So basically with mean and var, you have only two of the moments so unless your distribution is diatic, you cannot uniquely identify the distribution in question. Instead, you can find a family of distributions. All this assumes mean and var as the ONLY information you have, if you put some structural constraints, then the problem would be totally different. For example, if you know it the distribution is binomial or Poisson, then finding the exact distribution is straightforward.