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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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\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.