Suppose $X_1....X_n$ is a random sample from a Poi($\theta$) population. Find the best unbiased estimator of $\theta^2e^{-\theta}$
My attempt:
Let $\sum_1^nX_i=T$. We know $T$ is complete and sufficient. So we seek an unbiased estimator of $\theta^2e^{-\theta}$ then condition it on T then find the expected value.
An unbiased estimator of $\theta^2e^{-\theta}$ is $2 \chi_{[X_1=2]}$
We calculate $E(2\chi_{[X_1=2]}\mid T=t) = 2\Pr(X=2\mid T=t)$
By Bayes this is
$$2(tC2) \left( 1-\frac{1}{n}\right)^{t-2} \left( \frac{1}{n} \right)^2$$
I'm very unsure of this result. It matches none of my classmates, but I cannot see where I'm making an error. I also would be curious to see alternative approaches.