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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 8

Question

Math 6010 • Linear Models Fall 2017 • Question 8

Let \(X_1,X_2,\ldots ,X_n\) be iid with a Poisson\((\theta )\) distribution, so that their common probability mass function is \[ \mathbb {P}(X_i=k)=e^{-\theta }\frac {\theta ^k}{k!}, \qquad k=0,1,2,\ldots ,\ \theta >0. \] Find a sufficient statistic for \(\theta \).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid with a Poisson\((\theta)\)
distribution, so that their common probability mass function is
\[
  \Prob(X_i=k)=e^{-\theta}\frac{\theta^k}{k!}, \qquad
  k=0,1,2,\ldots,\ \theta>0.
\]
Find a sufficient statistic for \(\theta\).

Topic Tags

Estimator Distributions Measure-Theoretic Probability Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods