(i) What does it mean to say that a family $\{f(\cdot \mid \theta): \theta \in \Theta\}$ of densities is an exponential family?

Consider the family of densities on $(0, \infty)$ parametrised by the positive parameters $a, b$ and defined by

$$f(x \mid a, b)=\frac{a \exp \left(-(a-b x)^{2} / 2 x\right)}{\sqrt{2 \pi x^{3}}} \quad(x>0)$$

Prove that this family is an exponential family, and identify the natural parameters and the reference measure.

(ii) Let $\left(X_{1}, \ldots, X_{n}\right)$ be a sample drawn from the above distribution. Find the maximum-likelihood estimators of the parameters $(a, b)$. Find the Fisher information matrix of the family (in terms of the natural parameters). Briefly explain the significance of the Fisher information matrix in relation to unbiased estimation. Compute the mean of $X_{1}$ and of $X_{1}^{-1}$.