Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 5J / b / Solution

Independence gives the joint densityBy the factorization criterion, its dependence on the sample relevant to $\theta$ is only through ${\sum }_i{Y}_i$, equivalently through $\overline{Y}$. Thus $\overline{Y}$ is a sufficient statistic for $\theta$.