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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 6

Question

Math 6010 • Linear Models 2000 • Question 6

Let \(\mathbb {P}\{X=1\}=\mathbb {P}\{X=-1\}=1/2\), let \(X_1,X_2,X_3,\ldots \) be i.i.d. with common distribution that of \(X\), and put \(S_n=X_1+X_2+\cdots +X_n\). Define \(T=\min \{n\ge 0:S_n=-A\text { or }S_n=B\}\), where \(A\) and \(B\) are positive integers. Using the martingales of problem 5, find \(\mathbb {P}\{S_T=B\}\) and \(\mathbb {E}[T]\).

Raw LaTeX

Let \(\Prob\{X=1\}=\Prob\{X=-1\}=1/2\), let \(X_1,X_2,X_3,\ldots\) be
i.i.d. with common distribution that of \(X\), and put
\(S_n=X_1+X_2+\cdots+X_n\). Define
\(T=\min\{n\ge 0:S_n=-A\text{ or }S_n=B\}\), where \(A\) and \(B\) are positive
integers. Using the martingales of problem 5, find \(\Prob\{S_T=B\}\) and \(\E[T]\).

Topic Tags

Estimator Distributions Martingales and Supermartingales Multivariate Normal Distributions Uniform Integrability and Truncation Methods