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[Solution] Let X_1,...,X_n be i.i.d. observations from an exponential (λ) distribution, the p.d.f. of X is given by Use the moment-generating function

Question: Let ${{X}_{1}},...,{{X}_{n}}$ be i.i.d. observations from an exponential ( $\lambda $ ) distribution, the p.d.f. of X is given by

Use the moment-generating function of X to find the distribution of (10 points)
Show that T is sufficient for $\lambda $. (8 points)
It is easy to show that T is also complete for $\lambda $. Find the UMVUE of (6 points)

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[Solution] Let X_1,...,X_n be i.i.d. observations from an exponential (λ) distribution, the p.d.f. of X is given by Use the moment-generating function

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