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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2005 • Question 9

Question

Math 6010 • Linear Models 2005 • Question 9

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with a continuous probability distribution function \(F\) and \(Y_1,Y_2,\ldots ,Y_m\) be independent, identically distributed random variables with a continuous probability distribution function \(H\). The two samples are independent and \(F=H\). Let \[ T=\sum _{1\leq i\leq n}\sum _{1\leq j\leq m} I\{X_i\leq Y_j\}. \] Compute \(E T\) and \(\operatorname {var}T\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent, identically distributed random variables with a continuous probability distribution function \(F\) and \(Y_1,Y_2,\ldots,Y_m\) be independent, identically distributed random variables with a continuous probability distribution function \(H\). The two samples are independent and \(F=H\). Let
\[
T=\sum_{1\leq i\leq n}\sum_{1\leq j\leq m} I\{X_i\leq Y_j\}.
\]
Compute \(E T\) and \(\operatorname{var}T\).

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Topic Tags

Covariance and Independence Measure-Theoretic Probability Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods