Uniqueness of n and p in a discrete binomial distribution

Question

For a discrete random variable with binomial distribution, if $E(X)=9.6$ and $\text{Var}(X)=1.92$, will $n$ and $p$ have the unique values of $12$ and $.8$ or are other pairs of values possible?

Answer 1

If $p = 0$, then we will have $E[X] = \text{Var}[X] = 0$ so $n$ can be whatever we want.

However, if both the expectation and variance are nonzero, it is unique. Indeed, we know that $E[X] = np$ and $\text{Var}[X] = np(1-p)$. So we can immediately divide the two to get that $1 - p = \text{Var}[X] / E[X]$, and hence $p$ is uniquely determined. We can then substitute this value of $p$ into the first equation to conclude that $n$ is uniquely determined as well.