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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 7

Question

Math 6010 • Linear Models 2006 • Question 7

Let \(\{X_i\}_{i=1}^{\infty }\) denote a sequence of i.i.d. random variables with \(\mathbb {P}\{X_1=0\}=p\) and \(\mathbb {P}\{X_1=1\}=q:=1-p\). Consider the random sequence \(X_1X_2\cdots \), and compute \(\mathbb {E}[T]\), where \(T\) denotes the first time that the pattern “001” appears in the said sequence.

Raw LaTeX

Let \(\{X_i\}_{i=1}^{\infty}\) denote a sequence of i.i.d. random variables
with \(\Prob\{X_1=0\}=p\) and \(\Prob\{X_1=1\}=q:=1-p\). Consider the random sequence
\(X_1X_2\cdots\), and compute \(\E[T]\), where \(T\) denotes the first time that the pattern
``001'' appears in the said sequence.

Topic Tags

Laws of Large Numbers Measure-Theoretic Probability Uniform Integrability and Truncation Methods