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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?

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    Do you mean the mean of the random variable $X$? Then we have $\sum_1^5 p_i=1$, $\sum_1^5 ip_i=3.3$. Even given the constraints $p_i \ge 0$, this leaves a lot of freedom for the $p_i$. That would make it possible for you to specify other desired properties the $p_i$ should have. If the $3.3$ is instead a *sample mean*, we get a linear diophantine problem. If that is so, is the size of the sample known?2011-10-17
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    Simul-posted to MathOverflow, where it is well on its way to closure, http://mathoverflow.net/questions/78384/construct-a-distribution-for-discrete-random-variable-from-its-mean2011-10-17

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