I was asked to prove the following statement:
Let $X$ be a discrete random variable with $$P_\theta(X=x)=\frac{{\theta\choose x}{N-\theta\choose n-x}}{{N \choose n}},\ x = 0, 1, 2, \dots , \min(\theta, n),\ n − x \le N − \theta$$ where $n$ and $N$ are positive integers, $N \ge n$, and $\theta = 0, 1, \dots ,N$. Show that $X$ is complete. (Hint: Use proof by induction.)
I try to prove by introducing any function of $x$, saying $g(x)$ s.t. if $\mathbb{E}g(X)=0 \Rightarrow g(x)=0\ a.s.$
By expanding the expectation, it's equivalent to show that $$\sum\limits_{x=\max(0,n-N+\theta)}^{\min(\theta,n)}\frac{g(x)}{x!(\theta-x)!(n-x)!(N-\theta-n+x)!}=0,\ \ \forall\theta\in[N]$$
But then, I got stuck. Especially, I have no idea where the induction could be applied and on which parameter I should make an induction (N or n?) It would be great if any of you could give me the starting step for that induction. I would greatly appreciate it if you helped me finish the remaining proof.