Two teams play a series of games, stopping as soon as one of the team has three wins. Assume the games are independent and that the chance the first team wins is an unknown parameter $\displaystyle \theta \in (0,1) $. Let X denote the number of games the first team wins, and Y the number of games the other teams wins.

a) Find the joint mass function of X and Y.

b) Find a minimal sufficient statistic.

c) Is it complete?

My solution so far:

a)

I have the joint density being $\displaystyle P_ \theta (X=x,Y=y)= \theta ^x(1- \theta )^y $ where $\displaystyle x,y \in \{ 0,1,2,3 \} $ and $\displaystyle 2 < x+y<6 $

But then I'm pretty much stuck with finding minimal sufficient statistics here, any help, please? Thank you.