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Mathematics Qualifying Exam Question Bank

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Question

Math 6010 • Linear Models

Fall 2009 • Question 4

Question

Math 6010 • Linear Models Fall 2009 • Question 4

Let \(X_1,X_2,\ldots \) be an i.i.d. sequence with \(\mathbb {P}(X_1=1)=p\) and \(\mathbb {P}(X_1=0)=1-p\), where \(0<p<1\). For each \(m\geq 1\), define \[ N_m:=\min \{n\geq m:(X_{n-m+1},\ldots ,X_n)=(1,1,\ldots ,1)\} \] and show that \[ \mathbb {E}[N_m]=\frac {1}{p}+\frac {1}{p^2}+\cdots +\frac {1}{p^m}. \] Hint: Evaluate \(\mathbb {E}[N_m]\) by conditioning on \(N_{m-1}\).

Raw LaTeX

Let \(X_1,X_2,\ldots\) be an i.i.d. sequence with
\(\Prob(X_1=1)=p\) and \(\Prob(X_1=0)=1-p\), where \(0<p<1\). For each \(m\geq1\),
define
\[
  N_m:=\min\{n\geq m:(X_{n-m+1},\ldots,X_n)=(1,1,\ldots,1)\}
\]
and show that
\[
  \E[N_m]=\frac{1}{p}+\frac{1}{p^2}+\cdots+\frac{1}{p^m}.
\]
Hint: Evaluate \(\E[N_m]\) by conditioning on \(N_{m-1}\).

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Topic Tags

Laws of Large Numbers Uniform Integrability and Truncation Methods