Exercise 1.1

Let $Y_1,\dots ,Y_n$ be a random sample from a Poisson distribution $\mathit{Pois}(\lambda )$. In Example 1.6 we found that $T(Y)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^n{Y}_i$ is a sufficient statistic for $\lambda$ using the Fischer-Neyman factorization theorem.

1.

Obtain this result directly using the definition of a sufficient statistic. What is the distribution of $T(Y)$?

2.

Show that $T(Y)$ is a minimal sufficient statistic for $\lambda$.

3.

Show that $T(Y)$ is complete using

• the definition of complete statistic
• the properties of the exponential family of distributions.