Define the notion of exponential family $(E F)$, and show that, for data arising as a random sample of size $n$ from an exponential family, there exists a sufficient statistic whose dimension stays bounded as $n \rightarrow \infty$.

The log-density of a normal distribution $\mathcal{N}(\mu, v)$ can be expressed in the form

$$\log p(x \mid \boldsymbol{\phi})=\phi_{1} x+\phi_{2} x^{2}-k(\boldsymbol{\phi})$$

where $\phi=\left(\phi_{1}, \phi_{2}\right)$ is the value of an unknown parameter $\Phi=\left(\Phi_{1}, \Phi_{2}\right)$. Determine the function $k$, and the natural parameter-space $\mathbb{F}$. What is the mean-value parameter $\mathrm{H}=\left(\mathrm{H}_{1}, \mathrm{H}_{2}\right)$ in terms of $\Phi ?$

Determine the maximum likelihood estimator $\widehat{\Phi}_{1}$ of $\Phi_{1}$ based on a random sample $\left(X_{1}, \ldots, X_{n}\right)$, and give its asymptotic distribution for $n \rightarrow \infty$.

How would these answers be affected if the variance of $X$ were known to have value $v_{0}$ ?