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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2017 • Question 1

Question

Math 6010 • Linear Models Fall 2017 • Question 1

Let \(X_1,X_2,\ldots \) be independent random variables, and suppose that \(p_i:=\mathbb {P}\{X_i=0\}\) satisfies \(0<p_i<1\) for every \(i=1,2,\ldots \). For every \(n\geq 1\) let \[ N_n:=\sum _{i=1}^{n}\mathbf {1}_{\{X_i=0\}} \] denote the number of times the finite random sequence \(X_1,\ldots ,X_n\) enters zero. Prove that \[ \frac {N_n}{\sum _{i=1}^{n}p_i}\to 1 \qquad \text {in probability as }n\to \infty . \]

Raw LaTeX

Let \(X_1,X_2,\ldots\) be independent random variables, and suppose
that \(p_i:=\Prob\{X_i=0\}\) satisfies \(0<p_i<1\) for every \(i=1,2,\ldots\). For
every \(n\geq1\) let
\[
  N_n:=\sum_{i=1}^{n}\mathbf{1}_{\{X_i=0\}}
\]
denote the number of times the finite random sequence \(X_1,\ldots,X_n\) enters
zero. Prove that
\[
  \frac{N_n}{\sum_{i=1}^{n}p_i}\to1
  \qquad\text{in probability as }n\to\infty.
\]

Topic Tags

Classical Probability and Inclusion-Exclusion Covariance and Independence Measure-Theoretic Probability Uniform Integrability and Truncation Methods