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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2020 • Question 4

Question

Math 6010 • Linear Models Spring 2020 • Question 4

Let \(X\) and \(Y\) be any two random variables. Suppose \(\mathbb {E}[X^2]<\infty \). The conditional variance of \(X\) given \(Y\) is defined to be \[ \operatorname {Var}(X\mid Y)=\mathbb {E}\bigl [(X-\mathbb {E}[X\mid Y])^2\mid Y\bigr ]. \] Prove the conditional variance formula: \[ \operatorname {Var}(X)=\mathbb {E}[\operatorname {Var}(X\mid Y)]+\operatorname {Var}(\mathbb {E}[X\mid Y]). \]

Raw LaTeX

Let \(X\) and \(Y\) be any two random variables. Suppose
\(\E[X^2]<\infty\). The conditional variance of \(X\) given \(Y\) is defined to be
\[
  \Var(X\mid Y)=\E\bigl[(X-\E[X\mid Y])^2\mid Y\bigr].
\]
Prove the conditional variance formula:
\[
  \Var(X)=\E[\Var(X\mid Y)]+\Var(\E[X\mid Y]).
\]

Topic Tags

Analysis of Variance Conditional Expectation and Conditional Independence Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods