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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 1

Question

Math 6010 • Linear Models 1998 • Question 1

Suppose \(\{E_1,E_2,\ldots \}\) is a sequence of measurable (otherwise arbitrary) events. Suppose \(\{F_1,F_2,\ldots \}\) is another sequence of measurable events also totally independent of all of the \(E_i\)’s. Assume the following:

(a) \(\mathbb {P}(E_k\text {, infinitely often})=1\);

(b) there exists some \(p>0\), such that for all \(k\ge 1\), \(\mathbb {P}(F_k)\ge p\).

Prove: \(\mathbb {P}(E_k\cap F_k\text {, infinitely often})\ge p>0\). (10 points)

Raw LaTeX

Suppose \(\{E_1,E_2,\ldots\}\) is a sequence of measurable (otherwise
arbitrary) events. Suppose \(\{F_1,F_2,\ldots\}\) is another sequence of measurable
events also totally independent of all of the \(E_i\)'s. Assume the following:

(a) \(\Prob(E_k\text{, infinitely often})=1\);

(b) there exists some \(p>0\), such that for all \(k\ge 1\), \(\Prob(F_k)\ge p\).

Prove: \(\Prob(E_k\cap F_k\text{, infinitely often})\ge p>0\). (10 points)

Topic Tags

Borel-Cantelli Lemmas Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods