Let $X_1,X_2,...,X_n$ be i.i.d with Bernoulli distribution (so the p.m.f is $\theta^{x}(1-\theta)^{1-x})$.
I want to find the distribution of $\bar X$.
I calculated the moment generating function of $\bar X$ (by calculating $E[e^{\frac t n (X_1+...+X_n)}]$, and I found the M.G.F of $\bar X$ as $(1-\theta+\theta e^{\frac t n})^n.$ But I don't know the distribution. Is it $\text{Binom}(n,\theta)$, or $\text{Binom} (1,\theta),$ or $\text{Binom}(1,\frac \theta n)?$
As Ian points out in a comment, the distribution has no particular name -- it is a Binomial$(n,\theta)$ scaled by $n$. In particular, it it not itself a Binomial, and cannot be: it takes values of the form $\frac{m}{n}$ for $0\leq m \leq n$, i.e. is not integer-valued.