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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2008 • Question 5

Question

Math 6010 • Linear Models Spring 2008 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be i.i.d. positive random variables. Show that \[ \mathbb {E}\left [ \frac {\sum _{i=1}^m X_i}{\sum _{i=1}^n X_i} \right ] = \frac {m}{n},\qquad 1\leq m\leq n. \]

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be i.i.d. positive random variables. Show
that
\[
  \E\left[
    \frac{\sum_{i=1}^m X_i}{\sum_{i=1}^n X_i}
  \right]
  =
  \frac{m}{n},\qquad 1\leq m\leq n.
\]

\newpage

Topic Tags

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