Let $X_{1}, \ldots, X_{n}$ be independent, identically distributed $N\left(\mu, \sigma^{2}\right)$ random variables, where $\mu$ and $\sigma^{2}$ are unknown.

Derive the maximum likelihood estimators $\widehat{\mu}, \widehat{\sigma}^{2}$ of $\mu, \sigma^{2}$, based on $X_{1}, \ldots, X_{n}$. Show that $\widehat{\mu}$ and $\widehat{\sigma}^{2}$ are independent, and derive their distributions.

Suppose now it is intended to construct a "prediction interval" $I\left(X_{1}, \ldots, X_{n}\right)$ for a future, independent, $N\left(\mu, \sigma^{2}\right)$ random variable $X_{0}$. We require

$$P\left\{X_{0} \in I\left(X_{1}, \ldots, X_{n}\right)\right\}=1-\alpha$$

with the probability over the joint distribution of $X_{0}, X_{1}, \ldots, X_{n}$.

Let

$$I_{\gamma}\left(X_{1}, \ldots, X_{n}\right)=\left(\widehat{\mu}-\gamma \widehat{\sigma} \sqrt{1+\frac{1}{n}}, \widehat{\mu}+\gamma \widehat{\sigma} \sqrt{1+\frac{1}{n}}\right)$$

By considering the distribution of $\left(X_{0}-\widehat{\mu}\right) /\left(\widehat{\sigma} \sqrt{\frac{n+1}{n-1}}\right)$, find the value of $\gamma$ for which $P\left\{X_{0} \in I_{\gamma}\left(X_{1}, \ldots, X_{n}\right)\right\}=1-\alpha .$