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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 8

Question

Math 6010 • Linear Models Spring 2016 • Question 8

Let \(\{e_i,-\infty <i<\infty \}\) be independent and identically distributed random variables with \(\mathbb {E}[e_1]=0\) and \(\mathbb {E}[e_1^2]=\sigma ^2\) and \(\rho \) be a real number satisfying \(|\rho |<1\). Show that \[ Y=\sum _{i=0}^{\infty }\rho ^i e_i \] is finite with probability one. Compute \(\mathbb {E}[Y]\) and \(\mathbb {E}[Y^2]\). (You need to justify your steps when you compute these two expected values.)

Raw LaTeX

Let \(\{e_i,-\infty<i<\infty\}\) be independent and identically
distributed random variables with \(\E[e_1]=0\) and \(\E[e_1^2]=\sigma^2\) and
\(\rho\) be a real number satisfying \(|\rho|<1\). Show that
\[
  Y=\sum_{i=0}^{\infty}\rho^i e_i
\]
is finite with probability one. Compute \(\E[Y]\) and \(\E[Y^2]\). (You need to
justify your steps when you compute these two expected values.)

Topic Tags

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