If $X_1$ follows Binomial distribution with parameter $m$ and $p$ where $m$ is the number of trials and $p$ is the probability of success , that is , $X_1\sim B(m,p)$ and $X_2\sim B(n,p)$ then how can i prove that $$\hat p=\frac{X_1+X_2}{m+n}$$ is a sufficient statistic.
I know the way to find sufficient statistic by Factorization theorem. Also I have seen the link.
But i couldn't proceed to prove $\hat p$ is a sufficient statistic.
You have that $X_1+X_2 \sim B(m+n,p)$ (assuming that $X_1, X_2$ are independent). Then the result follows from the proof that the mean (or the sample proportion $\left(\hat{p}=\frac{X}{n}\right)$ is a sufficient statistic for the binomial distribution. This proof can be found as example in many books or online internet sources. For example see here.