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Statement of the lemma

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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

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

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

Then use integration by parts.