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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2011 • Question 5

Question

Math 6010 • Linear Models Spring 2011 • Question 5

Let \(X_1,\ldots ,X_n\) be independent identically distributed Poisson random variables for each \(n\) with parameter \(\lambda _n\).

(a) Show that \(X_1+\cdots +X_n\) is asymptotically normal if and only if \(n\lambda _n\to \infty \).

(b) Can you weaken the condition that the X’s are identically distributed for each \(n\)?

Raw LaTeX

Let \(X_1,\ldots,X_n\) be independent identically distributed Poisson
random variables for each \(n\) with parameter \(\lambda_n\).

\textbf{(a)} Show that \(X_1+\cdots+X_n\) is asymptotically normal if and only if
\(n\lambda_n\to\infty\).

\textbf{(b)} Can you weaken the condition that the X's are identically
distributed for each \(n\)?

Topic Tags

Covariance and Independence Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods