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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 9

Question

Math 6010 • Linear Models 2000 • Question 9

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed Poisson random variables with common probability mass function \[ p(k,\theta )=\frac {\theta ^k}{k!}e^{-\theta },\qquad k=0,1,\ldots \] Using the Cramer-Rao inequality show that the sample mean is a uniformly minimum variance unbiased estimator for \(\theta \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed Poisson random variables with common probability mass function
\[
p(k,\theta)=\frac{\theta^k}{k!}e^{-\theta},\qquad k=0,1,\ldots
\]
Using the Cramer-Rao inequality show that the sample mean is a uniformly minimum variance unbiased estimator for \(\theta\).

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Topic Tags

Estimator Distributions Measure-Theoretic Probability Probability Inequalities and Tail Bounds Sufficiency, Completeness, and UMVUE