Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2013 • Question 2

Question

Math 6010 • Linear Models Fall 2013 • Question 2

Let \(X_1,X_2,\ldots \) be i.i.d. with \(\mathbb {P}\{X_1=1\}=1/4\) and \(\mathbb {P}\{X_1=0\}=3/4\). Let \(N\) denote the smallest integer \(k\geq 2\) such that \(X_{k-1}=0,=X_k=1\); that is, \(N\) is the first time the pattern “01” occurs in the infinite sequence \(X_1,X_2,\ldots \). Compute \(\mathbb {E}(N)\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be i.i.d. with \(\Prob\{X_1=1\}=1/4\) and
\(\Prob\{X_1=0\}=3/4\). Let \(N\) denote the smallest integer \(k\geq2\) such that
\(X_{k-1}=0,=X_k=1\); that is, \(N\) is the first time the pattern ``01'' occurs in
the infinite sequence \(X_1,X_2,\ldots\). Compute \(\E(N)\).

\newpage

Topic Tags

Infinite Products and Multiplicative Martingales Laws of Large Numbers Uniform Integrability and Truncation Methods