Question: Consider a random sample of size n from a Poisson distribution, Xi ~ POI ( \(\mu \) ).

1. Find the CRLB for the variances of unbiased estimators of \(\mu \)
2. Find the CRLB for the variances of unbiased estimators of \(\theta ={{e}^{-\mu }}\).
3. Find a UMVUE of \(\mu \).
4. Find the MLE \(\bar{\theta }\) of \(\theta \).
5. Is \(\bar{\theta }\) an unbiased estimator of \(\theta \) ?
6. Is \(\bar{\theta }\) asymptotically unbiased?
7. Show that \(\tilde{\theta }={{\left( \frac{n-1}{n} \right)}^{\sum{{{X}_{i}}}}}\) is an unbiased estimator of theta.
   
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   Deliverable: Word Document
   
   
   

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