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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2014 • Question 10

Question

Math 6010 • Linear Models Fall 2014 • Question 10

Let \(Z=\operatorname {N}(0,1)\) denote a standard normal random variable. Suppose \(f:\mathbb {R}\to \mathbb {R}\) is continuously differentiable and there exists \(c>1\) such that \[ |f(x)|+|f'(x)|\leq c\exp (c|x|) \qquad \text {for all }x\in \mathbb {R}. \] Derive the following [Stein’s differential equation]: \[ \mathbb {E}[f'(Z)]=\mathbb {E}[Zf(Z)]. \]

Raw LaTeX

Let \(Z=\operatorname{N}(0,1)\) denote a standard normal random
variable. Suppose \(f:\R\to\R\) is continuously differentiable and there exists
\(c>1\) such that
\[
  |f(x)|+|f'(x)|\leq c\exp(c|x|)
  \qquad\text{for all }x\in\R.
\]
Derive the following [Stein's differential equation]:
\[
  \E[f'(Z)]=\E[Zf(Z)].
\]

Topic Tags

Central Limit Theorems Measure-Theoretic Probability Uniform Integrability and Truncation Methods