Question (Cramer-Rao inequality [5 marks]). Suppose that \( X_{1}, \cdots, X_{n} \) is a random sample from an exponential distribution with a parameter \( \lambda \). Prove that \( T=\bar{X} \) is the UMVU estimator of \( \lambda \) by applying the Cramer-Rao inequality for lower bounds.
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The PDF of an exponential distribution with parameter \(\lambda\) is given by: \[ f(x;\lambda) = \lambda e^{-\lambda x} \] for \( x \geq 0 \) and \(\lambda > 0\). Show more…
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