Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following a \(\mathrm{N}\left(\mu, \sigma^{2}ight)\) distribution. A saddlepoint expansion that approximates the density of \(n \bar{X}_{n}\) at a point \(x\) with relative error \(O\left(n^{-1}ight)\) as \(n ightarrow \infty\) was shown in Example 7.11 to have the form

\[\begin{aligned}

f_{n}(x)= & {\left[2 \pi n c^{\prime \prime}(\tilde{\lambda})ight]^{-1 / 2} \exp [n c(\tilde{\lambda})-\tilde{\lambda} x]\left[1+O\left(n^{-1}ight)ight] } \\

= & \left(2 \pi n \sigma^{2}ight)^{-1 / 2} \exp \left[\frac{1}{2} \sigma^{-2}\left(-n \mu^{2}+n^{-1} x^{2}ight)-x \sigma^{-2}\left(n^{-1} x-\muight)ight] \\

& \times\left[1+O\left(n^{-1}ight)ight] \\

= & \left(2 \pi n \sigma^{2}ight)^{-1 / 2} \exp \left[-\frac{1}{2} n^{-1} \sigma^{-2}(x-n \mu)^{2}ight]\left[1+O\left(n^{-1}ight)ight],

\end{aligned}\]

which has a leading term equal to a \(\mathrm{N}\left(n \mu, n \sigma^{2}ight)\) density, which matches the exact density of \(n \bar{X}_{n}\). Plot a \(\mathrm{N}\left(n \mu, n \sigma^{2}ight)\) density and the saddlepoint approximation for \(\mu=1, \sigma^{2}=1\), and \(n=2,5,10,25\) and 50. Discuss how well the saddlepoint approximation appears to be doing in each case.