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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 8

Question

Math 6010 • Linear Models 2006 • Question 8

Construct an example of a sequence \(\{A_n\}_{n=1}^{\infty }\) such that:

(a) (i) \(\sum _n\mathbb {P}(A_n)=\infty \); and (ii) only finitely-many of the \(A_n\)’s occur with positive probability.

(b) (i) \(\sum _n\mathbb {P}(A_n)=\infty \); and (ii) infinitely-many of the \(A_n\)’s occur with positive probability.

Raw LaTeX

Construct an example of a sequence \(\{A_n\}_{n=1}^{\infty}\) such that:

\textbf{(a)} \textbf{(i)} \(\sum_n\Prob(A_n)=\infty\); and \textbf{(ii)} only
finitely-many of the \(A_n\)'s occur with positive probability.

\textbf{(b)} \textbf{(i)} \(\sum_n\Prob(A_n)=\infty\); and \textbf{(ii)}
infinitely-many of the \(A_n\)'s occur with positive probability.

Topic Tags

Classical Probability and Inclusion-Exclusion Convergence in Probability and Almost Sure Convergence Measure-Theoretic Probability Uniform Integrability and Truncation Methods