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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 5

Question

Math 6010 • Linear Models 2004 • Question 5

Let \(\{X_n\}_{n=1}^{\infty }\) be i.i.d. random variables with distribution, \(\mathbb {P}\{X_1=2\}=\mathbb {P}\{X_1=-2\}=1/2\). Define \(S_n=\sum _{i=1}^{n}X_i\) and \(T=\inf \{n\ge 1:S_n\in \{-2,6\}\}\). Prove that \(T<\infty \) a.s., and then compute \(\mathbb {P}\{S_T=-2\}\).

Raw LaTeX

Let \(\{X_n\}_{n=1}^{\infty}\) be i.i.d. random variables with distribution,
\(\Prob\{X_1=2\}=\Prob\{X_1=-2\}=1/2\). Define \(S_n=\sum_{i=1}^{n}X_i\) and
\(T=\inf\{n\ge 1:S_n\in\{-2,6\}\}\). Prove that \(T<\infty\) a.s., and then compute
\(\Prob\{S_T=-2\}\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods