In following Maximum value of coefficient of variation for bounded data set I come up this question:
e.g., Say, $X$ can take integer values from $[0, 20]$ The mean of $X$ is known to be $0.005$
What is the maximum variance of $X$?
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3$\begingroup$ Is $X$ a random variable or a set of data? In the former case, it is well known (and easy to derive) that all the probability of $X$ must reside at the extreme values and the conditions of the problem give two simultaneous equations for those two probabilities that are easily solved. But if $X$ represents a dataset (as in the referenced question), this is a quadratic integer program whose solution varies according to the size of the dataset. There is no closed-form solution in that case. $\endgroup$ – whuber♦ Dec 13 '12 at 0:03
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Keeping in mind the comment above... if $X$ is a random variable, then the maximal variance distribution has probability $1-p$ at 0 and probability $p$ at 20; given the mean is 0.005, we can solve to get $p=1/4000$; this has variance $400 p (1-p) = 0.099975$.
The case where $X$ is a dataset doesn't have a closed-form solution, but is easy to solve for a particular case: begin with any configuration of the data that gives the right mean; then since $(a+1)^2+(b-1)^2 = x^2 + y^2 + 2(x-y+1)$ is positive if $x>y$, you can iteratively pick any pair of datapoints that are not at the endpoints, and move them each one unit apart. Given a particular sample size, you can work out what the distribution is from this.
