How to come up with a probability distribution knowing the mean value?

How to come up with a probability distribution knowing the mean value?

• I would like to know about some algorithms or techniques to find a discrete probability distribution knowing the mean value. Let's say given the mean=2.5. The probability distribution can be $x_1=2, P(x_1)=0.5$ and $x_2=3, P(x_2)=0.5$ $x_1, x_2,..., x_i$ should have discrete values in the solution.

• Answer:

  You may be interested in the idea of a http://en.wikipedia.org/wiki/Sufficient_statistic. What the commenters above are pointing out is that there are very few distributions for which just the mean is a sufficient statistic. One notable example which comes to mind is a Poisson distribution - the variance is equal to the expectation, so you could determine the entire distribution just from knowledge of the mean.