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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 6

Question

Math 6010 • Linear Models 2006 • Question 6

Suppose \(X\) and \(Y\) are two independent standard normal random variables.

(a) Prove that for all twice continuously differentiable functions \(f,g:\mathbb {R}\to \mathbb {R}\), \[ \operatorname {Cov}(f(X),g(X))=\int _0^1 \E \!\left [f'(X)g'\!\left (sX+(1-s^2)^{1/2}Y\right )\right ]\,ds. \]

(Hint: Check it first for \(f(x):=\exp (itx)\) and \(g(x):=\exp (itx)\).)

(b) Conclude the following “Poincaré inequality,” due to J. Nash (1958): \[ \Var (f(X))\le \E \!\left [(f'(X))^2\right ]. \]

Raw LaTeX

Suppose \(X\) and \(Y\) are two independent standard normal random
variables.

\textbf{(a)} Prove that for all twice continuously differentiable functions
\(f,g:\mathbb{R}\to\mathbb{R}\),
\[
  \Cov(f(X),g(X))=\int_0^1
  \E\!\left[f'(X)g'\!\left(sX+(1-s^2)^{1/2}Y\right)\right]\,ds.
\]

(Hint: Check it first for \(f(x):=\exp(itx)\) and \(g(x):=\exp(itx)\).)

\textbf{(b)} Conclude the following ``Poincar\'e inequality,'' due to J. Nash (1958):
\[
  \Var(f(X))\le \E\!\left[(f'(X))^2\right].
\]

Topic Tags

Central Limit Theorems Covariance and Independence Probability Inequalities and Tail Bounds Random Walks and Maximal Inequalities