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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2004 • Question 7

Question

Math 6010 • Linear Models 2004 • Question 7

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be independent, identically distributed random variables with a common continuous distribution function. Compute the expected value and the variance of \[ Z=\sum _{1\leq i\leq n}\sum _{1\leq j\leq n} I\{X_i\leq Y_j\}. \]

where \(I\{A\}\) denotes the indicator function of the event \(A\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) and \(Y_1,Y_2,\ldots,Y_m\) be independent, identically distributed random variables with a common continuous distribution function. Compute the expected value and the variance of
\[
Z=\sum_{1\leq i\leq n}\sum_{1\leq j\leq n} I\{X_i\leq Y_j\}.
\]

\newpage

where \(I\{A\}\) denotes the indicator function of the event \(A\).

Topic Tags

Analysis of Variance Covariance and Independence Estimator Distributions Uniform Integrability and Truncation Methods