Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2010 • Question 5

Question

Math 6010 • Linear Models Spring 2010 • Question 5

Fix \(n\geq 2\) and let \(X,Y_1,\ldots ,Y_n\) be jointly distributed random variables. We say that \(Y_1,\ldots ,Y_n\) are conditionally i.i.d. given \(X\) if \[ \mathbb {P}(Y_1\leq y_1,\ldots ,Y_n\leq y_n\mid X) =\mathbb {P}(Y_1\leq y_1\mid X)\cdots \mathbb {P}(Y_1\leq y_n\mid X) \]

for all \(y_1,\ldots ,y_n\). Show that, if \(Y_1,\ldots ,Y_n\) are conditionally i.i.d. given \(X\), then \[ \operatorname {Var}(Y_1+\cdots +Y_n)=n^2\operatorname {Var}(Y_1)-n(n-1)\mathbb {E}[\operatorname {Var}(Y_1\mid X)]. \]

Fix \(n\geq2\) and let \(X,Y_1,\ldots,Y_n\) be jointly distributed
random variables. We say that \(Y_1,\ldots,Y_n\) are conditionally i.i.d. given
\(X\) if
\[
  \Prob(Y_1\leq y_1,\ldots,Y_n\leq y_n\mid X)
  =\Prob(Y_1\leq y_1\mid X)\cdots\Prob(Y_1\leq y_n\mid X)
\]

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for all \(y_1,\ldots,y_n\). Show that, if \(Y_1,\ldots,Y_n\) are conditionally
i.i.d. given \(X\), then
\[
  \Var(Y_1+\cdots+Y_n)=n^2\Var(Y_1)-n(n-1)\E[\Var(Y_1\mid X)].
\]

Topic Tags

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