Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables following an EXPONENTIAL(1) density.

a. Prove that if \(\tilde{\lambda}=x^{-1}(x-n)\) then \(n c^{\prime}(\tilde{\lambda})=x\).

b. Prove that \(c(\tilde{\lambda})=\log \left(n^{-1} xight)\) and that \(c^{\prime \prime}(\tilde{\lambda})=n^{-2} x^{2}\).

c. Prove that the leading term of the saddlepoint expansion for \(f_{n}(x)\) is given by \((2 \pi)^{-1 / 2} n^{-n+1 / 2} x^{n-1} \exp (n-x)\).

d. The exact form for \(f_{n}(x)\) in this case is \(x^{n-1} \exp (-x) / \Gamma(n)\). Approximate \(\Gamma(n)\) using Theorem 1.8 (Stirling), and show that the resulting approximation matches the saddlepoint approximation.

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Theorem 7.8. Let {X} be a sequence of independent and identically dis- tributed random variables from a distribution F. Let Fn(x) = P[n/20 (Xn-