Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2016 • Question 7

Question

Math 6010 • Linear Models Fall 2016 • Question 7

Suppose \(X_1,X_2,\ldots \) are i.i.d. strictly positive random variables. Compute for all \(n,m,k\geq 1\), \[ \mathbb {E}\left ( \frac {X_1^k+\cdots +X_m^k}{X_1^k+\cdots +X_n^k} \right ). \] You might wish to start by calculating \(\mathbb {E}(X_1^k\mid X_1^k+\cdots +X_n^k)\).

Raw LaTeX

Suppose \(X_1,X_2,\ldots\) are i.i.d. strictly positive random
variables. Compute for all \(n,m,k\geq1\),
\[
  \E\left(
    \frac{X_1^k+\cdots+X_m^k}{X_1^k+\cdots+X_n^k}
  \right).
\]
You might wish to start by calculating
\(\E(X_1^k\mid X_1^k+\cdots+X_n^k)\).

Topic Tags

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