Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables where \(X_{n}\) has a density that is a mixture of two NORMAL densities of the form \(f(x)=\theta \phi(x)+(1-\theta) \phi(x-1)\) for all \(x \in \mathbb{R}\), where \(\theta \in[0,1]\).

a. Compute one- and two-term Edgeworth expansions for the density of \(n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight)\) where in this case \(\mu=E\left(X_{n}ight)\) and \(\sigma^{2}=V\left(X_{n}ight)\). What effect does the value of \(\theta\) have on the accuracy of the expansion? Is it possible to eliminate either the first or second term through a specific choice of \(\theta\) ?

b. Compute one- and two-term Edgeworth expansions for the distribution function of \(n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight)\).

c. Compute one- and two-term Cornish-Fisher expansions for the quantile function of \(n^{1 / 2} \sigma^{-1}\left(\bar{X}_{n}-\muight)\).