The variance of a binomial distribution is not adjustable. If you know its mean and the number of trials, the variance is also determined. I need a distribution that has an additional degree of freedom, where the variance and the mean are independent.
In other words, I need a discrete probability distribution with the following properties:
- Its support is the integers $0, 1, 2, \dots, N$, where $N$ is a non-negative integer (the number of trials, in analogy to the binomial distribution).
- The distribution is symmetric around $N/2$, which is also the mean.
- The variance is a free parameter that I can adjust (within a feasible range, see comments).
- For suitable parameters, the variance should go to zero if $N$ is even, or to $1/4$ if $N$ is odd (the variance of drawing one of $(N-1)/2$ or $(N+1)/2$ at random).
The binomial distribution with $p=1/2$ fulfills 1. and 2., but fails at 3 and 4, because the variance is a function of $p$ and $N$. Is there any other discrete probability distribution that meets these requirements?