A single observation is made from a poisson distribution with unknown mean $\lambda \geq 0$ However any value greater than 2 has been rounded down to 2. This we have the observed value of a single random variable X having distribution depending on $\lambda $ given by; $\ P(X=0) = e^{-\lambda}. P(X=1) = \lambda e^{-\lambda} P(X=2) = 1 - (1+\lambda)e^{-\lambda}$
Parameterise the distribution by $\ \theta = e^{-\lambda} \in (0,1] $ Show that there is a unique unbiased estimator of $\theta$.
So I parameterise it; $\ P(X=0) = \theta$ $\ P(X=1) = -\theta log\theta$ $\ P(X=2) = 1-(1-log\theta)\theta$
But I have no idea how to show there is a unique unbiased estimator. Also this is not a homework question, it is a practice paper question.