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Statement of the lemma

Suppose X is a normally distributed random variable with expectation μ and variance σ². Further suppose g is a function for which the two expectations E(g(X) (X − μ) ) and E( g ′(X) ) both exist (the existence of the expectation of any random variable is equivalent to the finiteness of the expectation of its absolute value). Then

Failed to parse (syntax error): {\displaystyle \begin{align*} E{g(X)(X-\theta)} &= \int_{-\infty}^{\infty} g(x)(x-\theta) f(x)dx\\ &=E{g'(X)} \end{align*}}

In general, suppose X and Y are jointly normally distributed. Then

$\operatorname {Cov} (g(X),Y)=E(g'(X))\operatorname {Cov} (X,Y).$

In order to prove the univariate version of this lemma, recall that the probability density function for the normal distribution with expectation 0 and variance 1 is

$\varphi (x)={1 \over {\sqrt {2\pi }}}e^{-x^{2}/2}$

and that for a normal distribution with expectation μ and variance σ² is

${1 \over \sigma }\varphi \left({x-\mu \over \sigma }\right).$

Then use integration by parts.