Let $X$ be a real random variable with the probability density function $x \mapsto \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ of the standard normal distribution. Establish the Stein identity $$\displaystyle {\bf E} X F(X) = {\bf E} F'(X)$$ whenever $F: {\bf R} \rightarrow {\bf R}$ is a continuously differentiable function with $F$ and $F'$ both of polynomial growth (i.e., there exist constants $C, A > 0$ such that $|F(t)|, |F'(t)| \leq C (1+|t|)^A$ for all $t \in {\bf R}$).

Homepage › Solution manuals › Terence Tao › Probability Theory › Exercise 1.37 (Stein identity)

Exercise 1.37 (Stein identity)

Let $X$ be a real random variable with the probability density function $x \mapsto \frac{1}{\sqrt{2 π}} e^{- x^{2} ∕ 2}$ of the standard normal distribution. Establish the Stein identity

EXF (X) = E F^{'} (X)

whenever $F : R \to R$ is a continuously differentiable function with $F$ and $F^{'}$ both of polynomial growth (i.e., there exist constants $C, A > 0$ such that $| F (t) |, | F^{'} (t) | \leq C {(1 + | t |)}^{A}$ for all $t \in R$ ).

Exercise 1.37 (Stein identity)

Add answer