Question: Let X ~ POI (mu), and let theta = P[X = 0] = \({{e}^{-\mu }}\).

1. Is \(\bar{\theta }\) = \({{e}^{-X}}\) an unbiased estimator of theta?
2. Show that \(\tilde{\theta }=u\left( X \right)\) is an unbiased estimator of theta, where u(0) = 1 and u(x) = 0 if x = 1,2,…..
3. Compare the MSEs of theta hat and theta tilde for estimating theta = e^-mu when mu = 1 and mu = 2.
   
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