I have found for a sample of binomially-distributed random variables with known numbers of trials (each r.v. has a unique number of trials) and an unknown common probability of success $p$ that their sufficient statistic for $p$ is $T(X_1, ... , X_n)$ is $\sum^n_{i=1}X_i$. I know the statistic is sufficient since the distribution belongs to the regular exponential family.
However, the next part of my tutorial question asks me to find the distribution of the statistic $T(X_1, ... , X_n)$.
Does anyone have any pointers for how to do this? I haven't been able to find any resources for finding the distribution of a statistic.