(a) Give the definitions of a sufficient and a minimal sufficient statistic $T$ for an unknown parameter $\theta$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be an independent sample from the geometric distribution with success probability $1 / \theta$ and mean $\theta>1$, i.e. with probability mass function

$$p(m)=\frac{1}{\theta}\left(1-\frac{1}{\theta}\right)^{m-1} \text { for } m=1,2, \ldots$$

Find a minimal sufficient statistic for $\theta$. Is your statistic a biased estimator of $\theta ?$

[You may use results from the course provided you state them clearly.]

(b) Define the bias of an estimator. What does it mean for an estimator to be unbiased?

Suppose that $Y$ has the truncated Poisson distribution with probability mass function

$$p(y)=\left(e^{\theta}-1\right)^{-1} \cdot \frac{\theta^{y}}{y !} \quad \text { for } y=1,2, \ldots$$

Show that the only unbiased estimator $T$ of $1-e^{-\theta}$ based on $Y$ is obtained by taking $T=0$ if $Y$ is odd and $T=2$ if $Y$ is even.

Is this a useful estimator? Justify your answer.