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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6040 • Probability

Fall 2024 • Question 4

Question

Math 6040 • Probability Fall 2024 • Question 4

Let \(A_{n}\) be a sequence of independent events with \(P(A_{n}) < 1\) for all \(n\). Show that \[ P\left (\bigcup _{n} A_{n}\right ) = 1 \quad \text {implies} \quad \sum _{n} P(A_{n}) = \infty . \]

Raw LaTeX

Let \(A_{n}\) be a sequence of independent events with \(P(A_{n}) < 1\) for all \(n\). Show that
\[
P\left(\bigcup_{n} A_{n}\right) = 1 \quad \text{implies} \quad \sum_{n} P(A_{n}) = \infty.
\]

Topic Tags

Classical Probability and Inclusion-Exclusion Uniform Integrability and Truncation Methods