Distribution of the Sum of Random Variables with a given distribution

Question

I need help finding a distribution in order to solve an exercise. We have the following problem:

Following this we want to calculate:

So I have found a complete sufficient statistic which is $\sum_{i=1}^n (Y_i)$. In order to find the $UMVUE$, I first need to find the distribution of $\sum_{i=1}^n (Y_i)$ | $β$. Can someone tell me how to do this??

I tried doing \begin{align} F_T(t) &= \mathrm P(T\leq t) = \mathrm P(\sum_{i=1}^n (Y_i)\leq t) \\ \end{align} How do I procede from here?

Thanks in advance for the help!