I wonder what are the ways for constructing a distribution over the values that a discrete random variable can take on given its mean.
For example, say a variable $x$ takes on an integer value from from $1$ to $5$, and now given the mean/expected value of $x$ over a population is $3.3$, what are the ways of constructing a distribution over $x$, which results the given mean, and what are the assumptions we need to make for each method? Thanks.
Edit (more context):
Say, a population of people are asked to each choose an integer from $1$ to $5$, and one of them (let's say $i$) chooses 4. In addition, $i$ estimates that the population mean is $3.3$. Now, I am interested in finding out, given the information provided by $i$, what can we say about $i$'s estimated distribution of the population's choices? In other words, by some reasonable assumptions or principle (e.g. maximum entropy), can we construct $i$'s estimated distribution?